This phase II trial studies how well taselisib (GDC-0032) works in treating patients with stage IV squamous cell lung cancer that has come back after previous treatment. This is a sub-study that includes all screened patients positive for the phosphoinositide 3-kinase (PI3K) biomarker. PI3K can cause tumor cells to grow more quickly. Taselisib may decrease the activity of PI3K and may be able to shrink tumors.