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In this paper, we investigate the group of endotrivial modules for certain$p$-groups. Such groups were already been computed by Carlson-Th\'evenaz usingthe theory of support varieties; however, we provide novel homotopical proofsof their results for cyclic $p$-groups, the quaternion group of order 8, andfor generalized quaternion groups using Galois descent and Picard spectralsequences, building on results of Mathew and Stojanoska. Our computationsprovide conceptual insights into the classical work of Carlson-Th\'evenaz.

Endotrivial modules for cyclic $p$-groups and generalized quaternion groups via Galois descent