Classifying abelian (but non-central) group extensions using homotopy theory

Question

Let $G$ be a group and let $A$ be an abelian group equipped with an action of $G$. Group extensions $$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$ inducing the given action of $G$ on $A$ are classified by elements of $H^2(G;A)$.

We can understand this topologically as follows. Associated to our group extension is a map of classifying spaces $B \Gamma \rightarrow BG$ with homotopy fiber $BA$. If the action of $G$ on $A$ is trivial (so the extension is central), then you can choose your models for $B\Gamma$ and $BG$ and $BA$ such that $BA$ is a topological group and $B\Gamma \rightarrow BG$ is a principal $BA$-fibration. These are classified by maps $BG \rightarrow B (BA)$, and since $B (BA)$ is a $K(A,2)$ this classification amounts to an element of $H^2(BG;A) = H^2(G;A)$.

The above fails for non-central extensions, and $B\Gamma \rightarrow BG$ is not a principal $BA$-bundle.

Question: Is there a similar homotopical description of the element of $H^2(G;A)$ associated to a non-central extension? I've seen various descriptions of classifying spaces for general fibrations, but I don't understand their obstruction theory well enough to understand how the local coefficients would enter the picture.

Answer 1

The short note Group Extensions and $H^3$ by P. J. Morandi works out the details of extensions by nonabelian groups. Any extension of $G$ by $A$ has an induced group homomorphism $\def\Out{{\sf Out}} ω\colon G→\Out(A)$. Conversely, given a group homomorphism $ω\colon G→\Out(A)$, we can construct a canonical cocycle with cohomology class $\def\H{{\sf H}}\def\Z{{\sf Z}}[c]∈\H^3(G,\Z(A))$, which vanishes if and only if $ω$ comes from a group extension.

If $[c]=0$, then trivializations of $c$ form a torsor over $\H^2(G,\Z(A))$ whose elements are in bijection with group extensions corresponding to $ω$.

This, in bundle-theoretic terms, a group homomorphism $ω\colon G→\Out(A)$ yields a bundle 2-gerbe over $\def\B{{\sf B}}\B G$ with structure group $\Z(A)$, whose trivializations are in bijection with group extensions corresponding to $ω$. Equivalently (using transgression), we can talk about a multiplicative gerbe over $G$ with structure group $\Z(A)$, whose (multiplicative) trivalizations are in bijection with group extensions corresponding to $ω$.