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Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution. For instance, let's take $G = \mathbb{Z}^2$, and "resolve": $$ 0 \to \...

By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...