# Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$

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).

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 $$G$$).