Quarterback 2b - Sequential Therapy With Reduced Dose Chemoradiotherapy for HPV Oropharynx Cancer

Inclusion Criteria: Participants must meet the following criteria to be eligible to participate in the study. Patients may enroll in the study if they meet all of the entry criteria, were candidates for induction chemotherapy regardless of HPV status and have to start therapy for logistic reasons prior to p16 and HPV testing. They will enter the experimental post-Induction portion of the study if the surgical specimens or biopsies are proven to be HPV+ on PCR testing. Patients who are HPV negative will be taken off study and treated with SOC radiotherapy and surgery: * Participants must have histologically or cytologically confirmed squamous cell carcinoma of the oropharynx, hypopharynx, supraglottic larynx, nasal cavity, unknown primary, or nasopharynx that is p16 and HPV positive. Tissue from the primary site or lymph node must be available for biomarker studies and for PCR testing. IHC must be performed in a lab verified by the central laboratory or the slides must be available for review by the central laboratory and PCR must be done in the central laboratory prior to radiotherapy. HPVPCR must be performed and results available for reduced dose therapy after induction. * Patients who are on the Quarterback Trial when Quarterback 2 is activated and who have been randomized to radiotherapy arm will be asked to transfer to this trial and receive the Quarterback 2 defined radiotherapy. * Stage 3 or 4 disease without evidence of distant metastases. * At least one clinically …