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

PRIMARY OBJECTIVES: I. To evaluate whether treatment with avelumab plus cetuximab prolongs progression free survival (PFS) compared to avelumab alone. SECONDARY OBJECTIVES: I. To evaluate the confirmed objective response rate of each treatment arm. II. To evaluate the clinical benefit rate of each treatment arm. III. To evaluate the PFS of cetuximab plus avelumab in patients that have progressed on single agent avelumab. IV. To evaluate the overall survival (OS) for each treatment arm. V. To evaluate toxicity across treatment arms of avelumab plus cetuximab and avelumab alone. OUTLINE: Patients are randomized to 1 of 2 arms. ARM I: Patients receive avelumab intravenously (IV) over 60 minutes on days 1 and 15. Treatment repeats every 28 days for up to 24 cycles in the absence of disease progression or unacceptable toxicity. Patients with avelumab failure will crossover to arm II. ARM II: Patients receive cetuximab IV over 1-2 hours on days 1, 8,15, and 22 and avelumab IV over 60 minutes on days 1 and 15. Treatment repeats every 28 days for up to 12 cycles for cetuximab and 24 cycles for avelumab in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up every 3 months until disease progression, then every 6 months for up to 2 years.