This phase II trial studies whether atezolizumab in combination with talazoparib works better than atezolizumab alone as maintenance therapy for patients with SLFN11-positive extensive-stage small cell lung cancer. Immunotherapy with monoclonal antibodies, such as atezolizumab, may help the body’s immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. PARPs are proteins that help repair damage to DNA, the genetic material that serves as the body’s instruction book. Changes (mutations) in DNA can cause tumor cells to grow quickly and out of control, but PARP inhibitors like talazoparib may keep PARP from working, so tumor cells can’t repair themselves, and they stop growing. Giving atezolizumab in combination with talazoparib may help lower the chance of extensive-stage small cell lung cancer growing and spreading compared to atezolizumab alone. The eligibility includes being at least 18 years old with additional requirements per study. Please contact a clinical trials coordinator for further questions.