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I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$ be a short exact sequence where $A$ maps to the center of $B$. Then there is a exact sequence $$1\rightarrow A^G\rightarrow B^G\rightarrow C^G\rightarrow H^1(G,A)\rightarrow H^1(G,B)\rightarrow H^1(G,C)\rightarrow H^2(G,A)$$ (where the last 4 things are just pointed sets, and the cohomology groups are the continuous ).

Gille-Szamuely's Central Simple Algebras and Galois Cohomology and Serre's Local fields (appendix) come very close. They describe the exact sequence when $G$ is finite (resp. an ordinary group, with no consideration of the topology).

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See Section 5.7 of Serre's Galois Cohomology. (In general Chapter 5 of this book is a fairly definitive reference for non-abelian cohomology, and he works with an arbitrary profinite group $G$).

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