Inclusion Criteria:

1. Histologically confirmed carcinoma of a primary tumor site and stage indicated in Table 3 of the protocol. 2. Tumor with HPV16 genotype as determined by testing performed in a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory. 3. HLA-A*02:01 allele determined by testing performed in a CLIA certified laboratory. Participants may be enrolled based on low resolution typing (i.e., HLA-A*02) but the HLA-A*02:01 allele type must be confirmed prior to apheresis. 4. Measurable disease per RECIST Criteria Version 1.1 or PERCIST. 5. Age > 18 years. 6. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at screening. 7. Negative pregnancy test for women under 55 and all women who have had a menstrual period in the last 12 months. A pregnancy tests is not required for women who have had a bilateral oophorectomy or hysterectomy. 8. Men and women of child-bearing potential must agree to use adequate contraception (i.e., intrauterine device, hormonal barrier method of birth control; abstinence; tubal ligation or vasectomy) prior to study entry and for four months after treatment. Should a women become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. 9. Seronegative for HIV antibody, hepatitis B surface antigen (sAg), and hepatitis C antibody. If a hepatitis C antibody test is positive, then testing for antigen by reverse transcription polymerase chain reaction (RT-PCR) must be negative. 10. Participants must have organ and marrow function as defined below: 1. Leukocytes > 3,000/Mantle cell lymphoma (mcL) 2. Absolute neutrophil count > 1,500/mcL 3. Platelets > 100,000/mcL 4. Hemoglobin > 9.0 g/dL 5. Total bilirubin within normal institutional limits except in participants with Gilbert's Syndrome who must have a total bilirubin < 3.0 mg/dL. 6. Serum aspartate aminotransferase (AST) (SGOT)/ alanine transaminase (ALT)(SGPT) < 2.5 x upper limit of normal (ULN) 7. Calculated creatinine clearance (CrCl) >50 mL/min/1.73 m2for participants with creatinine levels above institutional normal (by the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation). 8. international normalized ratio (INR) or activated partial thromboplastin time ( aPTT) ≤1.5 X ULN unless the subject is receiving anticoagulant therapy. Subjects on anticoagulant therapy must have a PT or aPTT within therapeutic range and no history of severe hemorrhage. 11. Participants must be able to understand and be willing to sign the written informed consent document. 12. Participants must agree to participate in protocol CINJ 192103 (Pro2021002307) for gene therapy long term follow up and in protocol Cancer Institute of New Jersey (CINJ) 192002 (Pro2021000281) for biospecimen collection study.Note: Patients may have undergone minor surgical procedures with the past three weeks, aslong as all toxicities have recovered to Grade 1 or less.

Exclusion Criteria: Subjects who meet any of the following criteria will be excluded from

participation in this study: 1. Prior systemic therapy or definitive chemoradiation for the cancer that is being treated on this protocol. 2. Current treatment with another investigational agent. 3. History of severe allergic reactions to compounds of similar chemical or biological composition to agents used in this study. 4. Uncontrolled intercurrent illness such as active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations at the time of treatment that would limit compliance with study requirements. 5. Subjects with HLA-A*02:01 damaging mutation or allele loss or other molecular resistance detected by research or clinical sequencing will not be eligible. 6. Documented LVEF of less than or equal to 45% tested. The following participants will undergo cardiac evaluations: 1. Clinically significant atrial and/or ventricular arrhythmias including but not limited to: atrial fibrillation, ventricular tachycardia, second or third degree heart block or 2. Age > 50 years old 7. Participants with baseline screening pulse oxygen level of <92% on room air will not be eligible. If the underlying cause of hypoxia improves, they may be reevaluated. 8. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with E7 TCR T cells, breastfeeding should be discontinued if the mother is treated with E7 TCR T cells. These potential risks may also apply to other agents used in this study. 9. Participants with a systemic immunodeficiency including acquired deficiency such as HIV or primary immunodeficiency such as Severe Combined Immunodeficiency Disease are ineligible. The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who have decreased immune competence may be less responsive to the treatment. 10. Participants on immunosuppressive drugs including corticosteroids unless meeting criteria outlined in Section 6.1 (Prohibited Medications). 11. Participants with potentially severe autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis, autoimmune pancreatitis, or systemic lupus erythematosus are not eligible. Patients with less severe autoimmune diseases such as hypothyroidism, vitiligo, and other minor autoimmune disorders are eligible. 12. Participants with prior or concurrent malignancy whose natural history or treatment is unlikely to interfere with the safety or efficacy assessments of the investigational regimen are eligible for this trial. Examples include, but are not limited to: 1. Carcinoma in situ 2. Cutaneous skin cancers requiring only local excision 3. Low grade non-muscle invasive bladder cancer 4. Low grade prostate cancer 13. Subjects who received a live vaccine within 30 days prior to enrollment are not eligible. 14. Determination by the Principal Investigator that participation is not in the best interest of the research subject or may jeopardize the safety of the subject or integrity of the clinical trial data.

Inclusion Criteria:

• Have histologically or cytologically confirmed recurrent/metastatic head and neck squamous cell carcinoma (R/M HNSCC) that is considered incurable by local therapies. Acceptable prior lines of therapy will be determined according to specific cohort 1, 2, 3A and 3B: (a) The eligible primary tumor locations are the oropharynx, oral cavity, hypopharynx, or larynx; (b) Any known p16 status of tumor must be negative (Note: All participants with an oropharyngeal tumor must have results of p16 status, per local testing); (c) Participants must provide local testing results of programmed cell death ligand 1 (PD-L1) status, if available; Cohort 4: (a) Patients must have primary tumor location in oropharynx. Unknown primary tumors are not included (b) Primary tumor must be HPV-positive, confirmed by positive p16 test or high-risk human papillomavirus (HPV) in-situ hybridization (ISH) in tissue (current or archival) (c) Participants must provide local testing results of PD-L1 status, if available; Cohort 5 (a) The eligible primary tumor locations are the oropharynx, oral cavity, hypopharynx, or larynx; (b) HPV status must be known (either positive or negative) for patients with primary tumor location in oropharynx with p16 test or high-risk HPV ISH in tissue; (c) Participants must provide local testing results of PD-L1 status
• Participants in Cohorts 1, 2, 3B, 4 and 5 must have measurable disease according to RECIST version 1.1. Participants in Cohort 3A must have evaluable disease (defined as having at least 1 non-target lesion according to RECIST version 1.1
• Toxicities from previous anticancer therapies should have resolved to baseline levels or to Grade 1 or less prior to the first dose of study treatment (except for alopecia or post-radiation skin changes [any grade], Grade less than or equal to [<=]2 peripheral neuropathy and Grade <=2 hypothyroidism stable on hormone replacement)
• Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1
• Participant must have adequate organ and bone marrow function as follows, without history of red blood cell transfusion, platelet transfusion, or use of granulocyte colony-stimulating factor within 7 days prior to the date of the laboratory test.Participants should have: a) Hemoglobin >=9 grams per deciliter (g/dL); b) Neutrophils>=1.5 x 10^3/mcg; c) Platelets >=100 x 10^3/mcg

Exclusion Criteria:

• Uncontrolled illness including any medical history or current (non-infectious) interstitial lung disease (ILD)/ pneumonitis/ pulmonary fibrosis, or where suspected ILD/pneumonitis/pulmonary fibrosis cannot be ruled out by imaging at screening
• Participant with untreated brain metastases leptomeningeal disease, or spinal cord compression not definitively treated with surgery or radiation
• Participant with a history of clinically significant cardiovascular disease
• Received prior chemotherapy, targeted cancer therapy, immunotherapy, or treatment with an investigational anticancer agent within 2 weeks or 4 half-lives, whichever is longer, before the first administration of study treatment. The maximum required washout is 28 days
• Received radiotherapy for palliative purposes within 7 days of the first administration of study treatment

Inclusion Criteria:

1. Histologically or cytologically confirmed metastatic or refractory/recurrent HPV-16+ cancer. 2. Tumor with HPV16 genotype as determined by testing performed in a CLIA certified laboratory. 3. HLA-A*02:01 allele as determined by testing performed in a Clinical Laboratory Improvement Amendments (CLIA) certified laboratory. Participants may be enrolled based on low resolution typing (i.e., HLA-A*02) but the HLA-A*02:01 allele type must be confirmed prior to apheresis. 4. Measurable disease as assessed by RECIST Criteria Version 1.1. 5. Age ≥ 18 years. 6. Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 at screening. 7. Must have received prior first line standard therapy or have declined standard therapy. 8. Standard treatment options for first and second-line therapy must be presented and formally declined (Appendix VII). 9. Patients with three or fewer brain metastases that have been treated with surgery or stereotactic radiosurgery are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month before protocol treatment. Patients must be fully recovered from surgery. 10. Negative pregnancy test for women under 55 and all women who have had a menstrual period in the last 12 months. A pregnancy tests is not required for women who have had a bilateral oophorectomy or hysterectomy. 11. Men and women of child-bearing potential must agree to use adequate contraception (i.e., intrauterine device, hormonal barrier method of birth control; abstinence; tubal ligation or vasectomy) prior to study entry and for four months after treatment. Should a women become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. 12. Seronegative for HIV antibody, hepatitis B antigen, and hepatitis C antibody. If a hepatitis C antibody test is positive, then testing for antigen by RT-PCR for Hepatitis C (HCV) RNA must be negative. 13. Participants must have organ and marrow function as defined below: 1. Leukocytes > 3,000/microliter (mcL) 2. Absolute neutrophil count > 1,500/mcL 3. Platelets > 100,000/mcL 4. Hemoglobin > 9.0 g/dL 5. Total bilirubin within normal institutional limits except in participants with Gilbert's Syndrome who must have a total bilirubin < 3.0 mg/dL. 6. Serum aspartate transferase (AST) (SGOT)/alanine transaminase (ALT) (SGPT) < 2.5 x upper limit of normal (ULN) 7. Calculated creatinine clearance (CrCl) >50 mL/min/1.73 m2for participants with creatinine levels above institutional normal (by the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation). 8. international normalized ratio (INR) or activated partial thromboplastin time ( aPTT) ≤1.5 X ULN unless the subject is receiving anticoagulant therapy. Subjects on anticoagulant therapy must have a PT or aPTT within therapeutic range and no history of severe hemorrhage. 14. More than four weeks must have elapsed since any prior systemic therapy at the time the patient receives the E7 TCR cells.. Adverse events from prior therapy must have resolved to ≤grade 1 according to CTCAE Version 5.0 or have demonstrated clinical stability for the protocol. 15. Participants must be able to understand and be willing to sign the written informed consent document. 16. Participants must agree to participate in protocol Cancer Institute of New Jersey (CINJ) 192103 (Pro2021002307) for gene therapy long term follow up and in protocol CINJ 192002 (Pro2021000281) for biospecimen collection study.Note: Participants may have undergone minor surgical procedures with the past threeweeks, as long as all toxicities have recovered to Grade 1 or less.

Exclusion Criteria: Subjects who meet any of the following criteria will be excluded from

participation in this study: 1. Uncontrolled intercurrent illness such as active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations at the time of treatment that would limit compliance with study requirements. 2. History of severe allergic reactions to compounds of similar chemical or biological composition to agents used in this study. 3. History of coronary revascularization or ischemic symptoms unless patient has a normal cardiac stress test. 4. Documented LVEF of less than or equal to 45% tested. The following participants will undergo cardiac evaluations: 1. Clinically significant atrial and/or ventricular arrhythmias including but not limited to: atrial fibrillation, ventricular tachycardia, second or third degree heart block or 2. Age > 50 years old 5. Participants with baseline screening pulse oxygen level of < 92% on room air will not be eligible. If the underlying cause of hypoxia improves, then they may be reevaluated. 6. Subjects with HLA-A*02:01 damaging mutation or allele loss or other molecular resistance detected by clinical or research genomic profiling will not be eligible. 7. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with E7 TCR T cells, breastfeeding should be discontinued if the mother is treated with E7 TCR T cells. These potential risks may also apply to other agents used in this study. 8. Participants with a systemic immunodeficiency including acquired deficiency such as HIV or primary immunodeficiency such as Severe Combined Immunodeficiency Disease are ineligible. The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who have decreased immune competence may be less responsive to the treatment. 9. Participants on immunosuppressive drugs including corticosteroids unless meeting criteria outlined in Section 6.1 (Prohibited Medications). 10. Participants with potentially severe autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis, autoimmune pancreatitis, or systemic lupus erythematosus are not eligible. Patients with less severe autoimmune diseases such as hypothyroidism, vitiligo, and other minor autoimmune disorders are eligible. 11. Participants with prior or concurrent malignancy whose natural history or treatment is unlikely to interfere with the safety or efficacy assessments of the investigational regimen are eligible for this trial. Examples include, but are not limited to: 1. Carcinoma in situ 2. Cutaneous skin cancers requiring only local excision 3. Low grade non-muscle invasive bladder cancer 4. Low grade prostate cancer Participants with prior or concurrent malignancy that do not meet the above criteria are excluded. 12. Subjects who received a live vaccine within 30 days prior to enrollment are not eligible. 13. Determination by the Principal Investigator that participation is not in the best interest of the research subject or may jeopardize the safety of the subject or integrity of the clinical trial data.

Inclusion Criteria:

• Patients must have pathologically confirmed HNSCC (including tumors of the oropharynx, hypopharynx, larynx, oral cavity, nasal cavity, maxillary and other paranasal sinuses, and unknown primary of the head and neck), with measurable disease as per RECIST 1.1
• Oropharyngeal and unknown primary squamous cell cancers must test for human papilloma virus (HPV), for example by p16 immunohistochemistry (IHC), in situ hybridization (ISH), or polymerase chain reaction (PCR). HPV testing is not required for other HNSCC primary tumor sites
• For the dose escalation phase only (not the expansion phase), patients with p16-positive tumors are eligible if clinical stage III (cT4 or cN3, M0) according to the American Joint Committee on Cancer (AJCC)/TNM Staging System, 8th edition (Ed.)
• For both the dose escalation and expansion phases, patients with p16-negative (or not tested) tumors are eligible if clinical stage III-IVB (locally advanced but non-metastatic) according to the AJCC/TNM Staging System, 8th Ed.
• Must be candidate for concurrent, definitive cisplatin and radiation therapy as judged by the treating physician
• Able to swallow tablets at the time of enrollment
• Age >= 18 years. Because no dosing or adverse event data are currently available on the use of ipatasertib in combination with chemoradiation in patients < 18 years of age, children are excluded from this study
• Eastern Cooperative Oncology Group (ECOG) performance status 0-1
• Life expectancy of greater than 3 months
• Absolute neutrophil count >= 1500/mcL
• Hemoglobin >= 9 g/dL
• Platelets >= 100,000/mcL
• Serum albumin >= 3 g/dL
• Total bilirubin =< 1.5 x institutional upper limit of normal (ULN)
• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 2.5 x institutional ULN / 2 x institutional ULN
• Alkaline phosphatase (ALP) =< 2.0 x institutional ULN
• Partial thromboplastin time (PTT) (or activated [a]PTT) and international normalized ratio (INR) =< 1.5 institutional ULN (except for patients receiving anticoagulation therapy)
• Creatinine clearance (CLcr) > 50 mL/min
• For this calculation, use the Cockroft-Gault formula
• Fasting glucose =< 150 mg/dL (8.3 mmol/L) and (when indicated) glycosylated hemoglobin (HbA1c ) =< 7.5% (58 mmol/mol)
• Human immunodeficiency virus (HIV)-infected patients are eligible if on effective anti-retroviral therapy with undetectable viral load within 6 months
• Patients with past hepatitis B virus (HBV) infection or resolved HBV infection (defined as having a negative hepatitis B virus surface antigen [HBsAg] test and a positive hepatitis B core antibody [HBcAb] test, accompanied by a negative HBV deoxyribonucleic acid [DNA] test) are eligible. Patients with chronic HBV infection are eligible if the HBV viral load is undetectable on suppressive therapy, if indicated. Patients undergoing current treatment with anti-viral therapy for HBV are ineligible
• Patients with a history of hepatitis C virus (HCV) infection are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA). Patients with HCV infection who are currently on treatment are eligible if they have an undetectable HCV viral load
• Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
• The effects of ipatasertib on the developing human fetus are unknown. For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods with a failure rate of < 1% per year during the treatment period and for at least 28 days after the last dose of ipatasertib and agreement to refrain from donating eggs during this same period. For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm during the treatment period and for 28 days after the last dose of ipatasertib
• Ability to understand and the willingness to sign a written informed consent document
• For the expansion cohort only, patients must agree to undergo mandatory on-treatment biopsies, and have tumors amenable to on-treatment biopsies. This is not applicable to the dose escalation cohort where no on-treatment biopsies are obtained

Exclusion Criteria:

• Primary tumor of nasopharynx, salivary, thyroid or parathyroid glands, or skin
• Distant metastases from the current HNSCC
• Prior treatment (e.g., chemotherapy, radiation, or definitive surgery) for the current locally advanced HNSCC is not permitted. Biopsies, including those performed under anesthesia, are not considered surgery. Patients who underwent prior definitive surgery alone for an early stage (T1-2N0) HNSCC which has now recurred with stage III-IVB disease at least 3 months after the initial surgery are eligible
• For patients with a prior history of another malignancy, no prior chemotherapy or radiation may have been administered within 6 weeks prior to study entry. Among patients who received prior radiation to the head and neck or adjacent anatomical site for another malignancy, there may be no overlap with current area to be irradiated
• Current use of any other investigational agents
• History of allergic reactions attributed to compounds of similar chemical or biologic composition to ipatasertib or other agents used in study
• Treatment with strong inhibitors or inducers of CYP3A4 or P-glycoprotein within 2 weeks or 5 drug-elimination half-lives, whichever is longer, prior to initiation of study drug. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product
• Patients with uncontrolled intercurrent illness, including active infection
• Pregnant women are excluded from this study because ipatasertib is an oral AKT inhibitor with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with ipatasertib, breastfeeding should be discontinued if the mother is treated with ipatasertib. These potential risks may also apply to other agents used in this study
• Patients with type I or type II diabetes mellitus requiring insulin at study entry. Patients with non-insulin dependent type II diabetes mellitus are eligible, as are patients who are on a stable dose of oral diabetes medication >= 4 weeks prior to initiation of study treatment. Patients with a history of diabetes mellitus, an abnormal fasting glucose level, or other signs or symptoms indicating diabetes mellitus, must meet the laboratory eligibility criteria for fasting blood glucose and hemoglobin A1c
• History of or active inflammatory bowel disease (e.g., Crohn's disease and ulcerative colitis) or active bowel inflammation (e.g., diverticulitis)
• History of malabsorption syndrome or other condition that would interfere with enteral absorption or results in the inability or unwillingness to swallow pills
• Lung disease: pneumonitis, interstitial lung disease, idiopathic pulmonary fibrosis, cystic fibrosis, aspergillosis, active tuberculosis, or history of opportunistic infections (pneumocystis pneumonia or cytomegalovirus pneumonia)
• Known clinically significant history of liver disease consistent with Child Pugh Class B or C, including active viral or other hepatitis (e.g., positive for hepatitis B surface antigen [HBsAg] or hepatitis C virus [HCV] antibody at screening), or cirrhosis
• Grade >= 2 uncontrolled or untreated hypercholesterolemia (cholesterol > 300 mg/dL or > 7.75 mmol/L) or hypertriglyceridemia (triglycerides > 300 mg/dL or > 3.42 mmol/L)

Inclusion Criteria:

• PRE-REGISTRATION (STEP 0)
• Pathologically proven diagnosis of squamous cell carcinoma (including variants such as verrucous carcinoma, spindle cell carcinoma, carcinoma not otherwise specified [NOS]) of the headeck (oral cavity, oropharynx, hypopharynx or larynx); pathologic stage III or IVA (American Joint Committee on Cancer [AJCC] 8): T3-T4a, N0-3, M0 or T1-T2, N1-3, M0
• Patient has undergone total resection of the primary tumor with curative intent
• NOTE: Patient is to be pre-registered to screening (Step 0) and tissue submitted to Foundation Medicine as soon as possible after surgery in order to meet the 8 week deadline to register the patient to Step 1 after surgery; full assay minimum turn-around time is 17-24 days
• For oropharynx primary tumors, the patient must have negative human papillomavirus (HPV) status of the tumor as determined by p16 protein expression using immunohistochemistry (IHC)
• Patients with, per the operative and/or pathology report, positive margin(s) (tumor present at the cut or inked edge of the tumor) which is not superceded by an additional margin of tumor-negative tissue, nodal extracapsular extension, and/or gross residual disease after surgery are not eligible
• A paraffin-embedded surgical tumor tissue specimen has been located is available for shipment to Foundation Medicine, Inc. following pre-registration
• NOTE: Complete the EA3132-specific FoundationOne requisition form
• Patients with a history of a curatively treated malignancy must be disease-free for at least two years except for carcinoma in situ of cervix and/or non-melanomatous skin cancer; patients must not have received chemotherapy or investigational therapy within two years of surgical resection of the primary tumor
• Patient must not have had previous irradiation to the head and neck that would result in overlap in radiation fields for the current disease
• Patients with recurrent disease or multiple primaries are ineligible
• RANDOMIZATION (STEP 1)
• NOTE: Patient must meet all eligibility criteria outlined in pre-registration; patient may not be randomized until site has been notified that the central determination of p53 mutation status of the surgical tumor tissue has been completed and site has been notified of assay completion
• Per the operative report, the gross total resection of the primary tumor with curative intent was completed within 8 weeks prior to randomization
• The patient must have the following assessments done =< 8 weeks prior to randomization:
• Examination by a head and neck surgeon
• Chest x-ray (or chest computed tomography [CT] scan or CT/positron emission tomography [PET] of the chest or magnetic resonance imaging [MRI]) to rule out distant metastatic disease
• Patient has Eastern Cooperative Oncology Group (ECOG) performance status 0-1 within 2 weeks prior to randomization
• Women must not be pregnant or breast-feeding; females of childbearing potential must have a blood or urine study within 2 weeks prior to randomization to rule out pregnancy; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)
• Women of childbearing potential and sexually active males must be strongly advised to use an accepted and effective method of contraception or to abstain from sexual intercourse for the duration of their participation in the study and until 60 days from the last study treatment
• Absolute neutrophil count >= 1,500/mm^3 within 4 weeks prior to randomization
• Platelets >= 100,000/mm^3 within 4 weeks prior to randomization
• Total bilirubin =< the upper limit of normal (ULN) within 4 weeks prior to randomization
• Calculated creatinine clearance must be > 60 ml/min using the Cockcroft-Gault formula within 4 weeks prior to randomization
• Patient must not have an intercurrent illness likely to interfere with protocol therapy

Inclusion Criteria:

• PHASE II INCLUSION CRITERIA (COMPLETE AS OF 20-MAR-2020)
• Pathologically (histologically or cytologically) proven diagnosis of head and neck squamous cell carcinoma (HNSCC) involving the oral cavity (excluding lips), oropharynx (p16 negative), larynx, or hypopharynx
• Patients must have undergone gross total surgical resection of high-risk oral cavity, oropharynx (p16 negative), larynx, or hypopharynx within 63 days prior to registration; Note: patients may have biopsy under general anesthesia in an operating room followed by definitive ablative cancer surgery representing gross total resection; the gross total resection has to be done within 63 days prior to registration; if, however, patients have ablative resection but shortly recur or are determined to have persisting disease requiring re-resection to achieve gross total resection, then the patient is not eligible
• Patients must have at least 1 of the following high-risk pathologic features: extracapsular nodal extension or invasive cancer at the primary tumor resection margin (tumor on ink)
• Pathologic stage III or IV HNSCC, including no distant metastases, based upon the following minimum diagnostic workup:
• General history and physical examination by a radiation oncologist and/or medical oncologist within 84 days prior to registration;
• Examination by an ear nose throat (ENT) or head & neck surgeon prior to surgery; a laryngopharyngoscopy (mirror and/or fiber optic and/or direct procedure), if appropriate, is recommended but not required; intra-operative examination is acceptable documentation
• Pre-operative (op) Imaging of the head and neck: A neck computed tomography (CT) (with contrast) or CT/positron emission tomography (PET) (with contrast) and/or an magnetic resonance imaging (MRI) of the neck (T1 with gadolinium and T2) within 84 days prior to surgery; Note: this imaging data (diagnostic pre-operative scan showing gross disease) is to be submitted in Digital Imaging and Communications in Medicine (DICOM) format via TRIAD; the report is to be uploaded into Rave
• Chest CT scan (with or without contrast) or CT/PET that includes the chest (with or without contrast) either within 84 days prior to surgery or within 120 days prior to registration; Note: if the CT/PET with or without contrast is done within 84 days prior to surgery, it fulfills the chest imaging requirement
• Zubrod performance status of 0-1 within 14 days prior to registration
• Age >= 18
• Absolute granulocyte count (AGC) >= 1,500 cells/mm^3 (obtained within 14 days prior to registration on study)
• Platelets >= 100,000 cells/mm^3 (obtained within 14 days prior to registration on study)
• Hemoglobin >= 8.0 g/dl (Note: the use of transfusion or other intervention to achieve hemoglobin [Hgb] >= 8.0 g/dl is acceptable)
• Total bilirubin < 2 x institutional upper limit of normal (ULN) within 14 days prior to registration
• Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) < 3 x institutional ULN within 14 days prior to registration
• Serum creatinine institutional ULN within 14 days prior to registration or; creatinine clearance (CC) >= 50 ml/min within 14 days prior to registration determined by 24-hour collection or estimated by Cockcroft-Gault formula
• Negative urine or serum pregnancy test within 14 days prior to registration for women of childbearing potential
• The following assessments are required within 14 days prior to registration: sodium (Na), potassium (K), chloride (Cl), glucose, calcium (Ca), magnesium (Mg), and albumin; Note: patients with an initial magnesium < 0.5 mmol/L (1.2 mg/dl) may receive corrective magnesium supplementation but should continue to receive either prophylactic weekly infusion of magnesium and/or oral magnesium supplementation (e.g., magnesium oxide) at the investigator's discretion
• Patients with feeding tubes are eligible for the study
• Women of childbearing potential and male participants who are sexually active must agree to use a medically effective means of birth control
• Patient must provide study specific informed consent prior to study entry, including consent for mandatory tissue submission for epidermal growth factor receptor (EGFR) analysis and for oropharyngeal cancer patients, human papilloma virus (HPV) analysis
• PHASE III: Pathologically (histologically or cytologically) proven diagnosis of head and neck squamous cell carcinoma (HNSCC) involving the oral cavity (excluding lips), oropharynx (p16 negative), larynx, or hypopharynx
• PHASE III: Patients with oropharyngeal cancer must have p16-negative based on central review prior to Step 2 registration. All patients with oropharyngeal primary must consent for mandatory tissue submission for central p16 confirmation
• PHASE III: Patients must have undergone gross total surgical resection of high-risk oral cavity, oropharynx (p16 negative), larynx, or hypopharynx within 63 days prior to registration
• Note: Patients may have biopsy under general anesthesia in an operating room followed by definitive ablative cancer surgery representing gross total resection. The gross total resection has to be done within 63 days prior to registration. If, however, patients have ablative resection but shortly recur or are determined to have persisting disease requiring re-resection to achieve gross total resection, then the patient is not eligible
• PHASE III: Patients must have at least 1 of the following high-risk pathologic features: extracapsular nodal extension or invasive cancer at the primary tumor resection margin (tumor on ink or tumor in a final separately submitted margin)
• PHASE III: Pathologic stage III or IV HNSCC (American Joint Committee on Cancer [AJCC] 7th edition), including no distant metastases, based upon the following minimum diagnostic workup:
• General history and physical examination by a radiation oncologist or medical oncologist within 84 days prior to registration;
• Examination by an ENT or head & neck surgeon prior to surgery; a laryngopharyngoscopy (mirror or fiberoptic or direct procedure), if appropriate, is recommended but not required. Intra-operative examination is acceptable documentation.
• Pre-op Imaging of the head and neck: A neck CT (with contrast and of diagnostic quality) or PET/CT (with contrast and of diagnostic quality) and/or an MRI of the neck of diagnostic quality (T1 with gadolinium and T2) within 84 days prior to surgery; Note: this imaging data (diagnostic pre-operative scan showing gross disease) is to be submitted in DICOM format via TRIAD. The report is to be uploaded into Rave.
• Chest CT scan (with or without contrast) or PET/CT that includes the chest (with or without contrast) either within 84 days prior to surgery or within 120 days prior to registration; Note: If the PET/CT with or without contrast is done within 84 days prior to surgery, it fulfills the chest imaging requirement
• PHASE III: Zubrod performance status of 0-1 within 14 days prior to registration
• PHASE III: Age >= 18
• PHASE III: Leukocytes >= 2,500 cells/mm^3 (obtained within 14 days prior to registration on study)
• PHASE III: Absolute neutrophil count (ANC) >= 1,500 cells/mm^3 (obtained within 14 days prior to registration on study)
• PHASE III: Platelets >= 100,000 cells/mm^3 (obtained within 14 days prior to registration on study)
• PHASE III: Hemoglobin >= 8.0 g/dL (Note: The use of transfusion or other intervention to achieve Hgb >= 8.0 g/dL is acceptable) (obtained within 14 days prior to registration on study)
• PHASE III: Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =< 3 x institutional ULN may be enrolled) (within 14 days prior to registration)
• PHASE III: AST or ALT =< 3 x institutional ULN (within 14 days prior to registration)
• PHASE III: Alkaline phosphatase =< 2.5 x institutional ULN (within 14 days prior to registration)
• PHASE III: Creatinine clearance (CrCl) >= 50 mL/min within 14 days prior to registration determined by 24-hour collection or estimated by Cockcroft-Gault formula
• PHASE III: Patients with feeding tubes are eligible for the study
• PHASE III: Negative urine or serum pregnancy test within 14 days prior to registration for women of childbearing potential
• PHASE III: All patients must provide study specific informed consent prior to study entry
• PHASE III: Patients positive for human immunodeficiency virus (HIV) are allowed on study, but HIV-positive patients must have:
• A stable regimen of highly active anti-retroviral therapy (HAART);
• No requirement for concurrent antibiotics or antifungal agents for the prevention of opportunistic infections;
• A CD4 count above 250 cells/mcL and an undetectable HIV viral load on standard polymerase chain reaction (PCR)-based tests

Exclusion Criteria:

• PHASE II EXCLUSION CRITERIA (COMPLETE AS OF 20-MAR-2020)
• Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 1095 days (3 years); noninvasive cancers (for example, carcinoma in situ of the breast, oral cavity, or cervix are all permissible) are permitted even if diagnosed and treated < 3 years ago
• Patients with simultaneous primaries or bilateral tumors are excluded, with the exception of patients with bilateral tonsil cancers or patients with T1-2, N0, M0 resected differentiated thyroid carcinoma, who are eligible
• Prior systemic chemotherapy or anti-epidermal growth factor (EGF) therapy for the study cancer; note that prior chemotherapy for a different cancer is allowable
• Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
• Severe, active co-morbidity, defined as follows:
• Unstable angina and/or congestive heart failure requiring hospitalization within 6 months prior to registration
• Transmural myocardial infarction within 6 months prior to registration
• Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration
• Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration
• Idiopathic pulmonary fibrosis or other severe interstitial lung disease that requires oxygen therapy or is thought to require oxygen therapy within 1 year prior to registration
• Hepatic insufficiency resulting in clinical jaundice and/or coagulation defects; note, however, that laboratory tests for coagulation parameters are not required for entry into this protocol
• Acquired immune deficiency syndrome (AIDS) based upon current Centers for Disease and Control and Prevention (CDC) definition; note: human immunodeficiency virus (HIV) testing is not required for entry into this protocol; the need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive; protocol-specific requirements may also exclude immuno-compromised patients.
• Grade 3-4 electrolyte abnormalities (Common Terminology Criteria for Adverse Events [CTCAE], version [v.] 4):
• Serum calcium (ionized or adjusted for albumin) < 7 mg/dl (1.75 mmol/L) or > 12.5 mg/dl (> 3.1 mmol/L) despite intervention to normalize levels
• Glucose < 40 mg/dl (< 2.2 mmol/L) or > 250 mg/dl (> 14 mmol/L)
• Magnesium < 0.9 mg/dl (< 0.4 mmol/L) or > 3 mg/dl (> 1.23 mmol/L) despite intervention to normalize levels
• Potassium < 3.0 mmol/L or > 6 mmol/L despite intervention to normalize levels
• Sodium < 130 mmol/L or > 155 mmol/L despite intervention to normalize levels
• Pregnancy or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception; this exclusion is necessary because the treatment involved in this study may be significantly teratogenic
• Prior allergic reaction to cetuximab
• PHASE III: Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 1095 days (3 years) with the following exceptions: T1-2, N0, M0 resected differentiated thyroid carcinoma; Note that noninvasive cancers (For example, carcinoma in situ of the breast, oral cavity, or cervix) are permitted even if diagnosed and treated < 3 years ago
• PHASE III: Patients with simultaneous primaries or bilateral tumors are excluded, with the exception of patients with bilateral tonsil cancers or patients with T1-2, N0, M0 resected differentiated thyroid carcinoma, who are eligible
• PHASE III: Prior systemic therapy, including cytotoxic chemotherapy, biologic/targeted therapy (such as anti-EGF therapy), or immune therapy for the study cancer; note that prior chemotherapy for a different cancer is allowable, however, a prior anti-PD-1, anti-PD-L1, or anti-PD-L2 agent is not permitted
• PHASE III: Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
• PHASE III: Severe, active co-morbidity, defined as follows:
• Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification; to be eligible for this trial, patients should be class 2B or better within 6 months prior to registration
• Transmural myocardial infarction within 6 months prior to registration;
• Severe infections within 4 weeks prior to registration including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia;
• Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration; Note: Patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease exacerbation) are eligible.
• Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration;
• History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan. History of radiation pneumonitis in a prior radiation field (fibrosis) is permitted, provided that field does not overlap with the planned radiation field for the study cancer;
• Patients with active tuberculosis (TB) are excluded;
• Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease;
• Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen [HBsAg] test and a positive anti-HBc [antibody to hepatitis B core antigen] antibody test) are eligible.
• Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV RNA.
• History of allogeneic bone marrow transplantation or solid organ transplantation.
• A diagnosis of immunodeficiency:
• Acquired immune deficiency syndrome (AIDS) based upon current CDC definition; note: HIV testing is not required for entry into this protocol; the need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive.
• Is receiving treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor [anti-TNF] agents) within 2 weeks prior to registration.
• Note: Patients who have received acute, low dose, systemic immunosuppressant medications (e.g., a one-time dose of dexamethasone for nausea) may be enrolled.
• Note: The use of inhaled corticosteroids and mineralocorticoids (e.g., fludrocortisone) for patients with orthostatic hypotension or adrenocortical insufficiency is allowed.
• History or risk of autoimmune disease, including, but not limited to, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Guillain-Barré syndrome, multiple sclerosis, vasculitis, or glomerulonephritis.
• Patients with a history of autoimmune hypothyroidism who are asymptomatic and/or are on a stable dose of thyroid replacement hormone are eligible.
• Patients with controlled Type 1 diabetes mellitus on a stable insulin regimen are eligible.
• Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions:
• Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations
• Rash must cover less than 10% of body surface area (BSA)
• Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, flucinolone 0.01%, desonide 0.05%, aclometasone dipropionate 0.05%)
• No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation [PUVA], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids)
• PHASE III: Grade 3-4 electrolyte abnormalities (CTCAE, v. 4) within 14 days prior to registration:
• Serum calcium (ionized or adjusted for albumin) < 7 mg/dL (1.75 mmol/L) or > 12.5 mg/dL (> 3.1 mmol/L) despite intervention to normalize levels;
• Glucose < 40 mg/dL (< 2.2 mmol/L) or > 250 mg/dL (> 14 mmol/L);
• Magnesium < 0.9 mg/dL (< 0.4 mmol/L) or > 3 mg/dL (> 1.23 mmol/L) despite intervention to normalize levels;
• Potassium < 3.0 mmol/L or > 6 mmol/L despite intervention to normalize levels;
• Sodium < 130 mmol/L or > 155 mmol/L despite intervention to normalize levels.
• PHASE III: Pregnancy or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception for up to 5 months from last study treatment; this exclusion is necessary because the treatment involved in this study may be significantly teratogenic. Women who are breastfeeding and unwilling to discontinue are also excluded
• PHASE III: History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins
• PHASE III: Patients taking bisphosphonate therapy for symptomatic hypercalcemia. Use of bisphosphonate therapy for other non-oncologic reasons (e.g., osteoporosis) is allowed
• PHASE III: Patients requiring treatment with a RANKL inhibitor (e.g. denosumab) for non-oncologic reasons who cannot discontinue it before registration
• PHASE III: Patients with known distant metastatic disease are excluded
• PHASE III: Known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies
• PHASE III: Major surgical procedure within 28 days prior to registration or anticipation of need for a major surgical procedure during the course of the study
• PHASE III: Administration of a live, attenuated vaccine within 4 weeks prior to registration or anticipation that such a live, attenuated vaccine will be required during the study and for patients receiving atezolizumab, up to 5 months after the last dose of atezolizumab.
• Influenza vaccination should be given during influenza season only (approximately October to