In an older question I wrote this:
It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extension of $G$ by $A$ is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation $\bullet$ is
$$(a_1,g_1)\bullet(a_2,g_2) = (a_1+\varphi_{g_1}(a_2)+f(g_1,g_2),\,g_1g_2)\tag{1}$$
where
- $\varphi:(A\times G)\to A: (a,g)\mapsto \varphi_g(a)$ is a group action of $G$ on $A$
- $f: G\times G\to A$ is a cocycle, i.e. satisfies $f(g_1,\,g_2g_3)+\varphi_{g_1}(f(g_2,\,g_3)) = f(g_1g_2,\,g_3)+f(g_1,\,g_2)$.
I've deduced (1) from Shafarevich's Basic Notions of Algebra and from this Wikpedia article but I dind't see this formula explicitly written nowhere. Does anybody know an explicit reference for this very formula?