Avelumab With or Without Cetuximab in Treating Patients With Advanced Skin Squamous Cell Cancer

Inclusion Criteria: * PRE-REGISTRATION ELIGIBILITY CRITERIA * Provide adequate tissue for PD-L1 testing \* Fresh tissue or archival tissue can be used. Sample must be at least core needle biopsy. Fine needle aspiration is not adequate. This specimen submission is mandatory prior to registration as results will be used for stratification * REGISTRATION ELIGIBILITY CRITERIA * • Biopsy-proven advanced cutaneous squamous cell carcinoma. Advanced disease is defined as either metastatic cutaneous squamous cell carcinoma or locally advanced cutaneous squamous cell carcinoma not amenable to curative surgical resection, or the patient declines surgical resection * The patient must have at least one lesion that is measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * If patient received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy * This study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown. Therefore, for women of childbearing potential only, a negative urine or serum pregnancy test done =\< 7 days prior to registration is required * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * If human immunodeficiency virus (HIV) positive the HIV viral load must be \< 200 copies/mL and CD4 count \> 200. If an HIV positive patient is on highly active antiretroviral t…