I do not really think it fits MO, but I posted it in MathStackExchange with little success, so...

Assume that we have a short exact sequence of, say, Lie groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$, where $A$ is abelian and ( probably it is necessary but I am not sure) $A$ maps to the center of $B$, and a topological space $X$. Then there is an associated (semi)long exact sequence of Cech cohomology $sets$: $*\rightarrow H^0(X,A)\rightarrow...\rightarrow H^1(X,C)\rightarrow H^2(X,A).$

My question is how to prove that naturally defined last differential $H^1(X,C)\rightarrow H^2(X,A)$ actually produces a cocycle, and so an element in $H^2(X,A)$.

When I tried to verify the cocycle condition, I certainly used that $A$ is commutative, to some extent used that $A$ maps to the center of $B$, but still was not able to finish this quite tricky caclulation due to non-commutativity of $B$.

I would appreciate the actual calculation or any reference where it is done.

A General Theory of Fibre Spaces with Structure Sheaf, 5.7, andSur quelques points d'algèbre homologique, Proposition 3.4.2. In the first paper he uses Cech cohomology and needs $X$ to be paracompact; this hypothesis is removed in the second paper by using "true" cohomology. $\endgroup$3more comments