I'm looking for a reference for the following "well known" result
Let $C$ be an abelian group and $G$ a finite group, and let $$0 \rightarrow C \rightarrow W \rightarrow G \rightarrow 0$$ be a group extension of $G$ by $C$ corresponding to the fundamental class in $H^2(C,G)$. Now the result I have says that if we have a homomorphism $f$ from $C$ to some abelian group $A$ and an exact sequence of groups $$0 \rightarrow A \rightarrow A \rtimes G \rightarrow G \rightarrow 0, $$ then you can extend $f$ to some map $$f': W \rightarrow A' \rtimes G$$ such that both these sequences fit into a commutative diagram with the first vertical arrow given by $f$ and the third given by the identity map if and only if $f$ is $G$ invariant and it "sends" (I don't quite get what this means) the fundamental class in $H^2(C,G)$ to zero.
The place I got this result from just gives a reference to Serre's Local fields but not a page number or anything that might narrow where this result is. I can imagine it is pretty standard but I cant seem to find anything of this sort.
Thank you
We had better suppose that $A$ is abelian, and so a $G$-module; otherwise I'm not sure the question makes sense (because the $H^2$ with coeffs. in $A$ would not be defined).
Group cohom. is functorial in the coefficients, so $C \to A$ induces $f_*: H^2(G,C) \to H^2(G,A)$. This latter map has an interpretation in terms of extensions: it sends the extension $0 \to C \to W \to G \to 0$ to the extension $0 \to A \to W' \to G \to 0$ obtained by "pushing out" along the morphism $C \to A$. One way to think about $W'$ is that it is the unique extension of $G$ by $A$ compatible with given action of $G$ on $A$, and which receives a map from $W$ that restricts to $f$ on $C$ and induces the identity on $G$.
Now by definition the $0$ class in $H^2$ corresponds to the split extension, i.e. the semi-direct product. Thus $f_*$ sends the class of $W$ (an element of $H^2G,C)$) to zero in $H^2(G,A)$ if and only if $W'$ is a semidirect product.
Now if you think about the above description of the push-out, you will see that $W'$ equals the semidirect product if and only if $W$ admits a map to the semidirect product, compatibly with the given map $f: C \to A$ and the identity on $G$.