H^2 classifies extensions with fixed abelian kernel action

For each extension under consideration, apply [A1] to the nonempty fibres of its quotient map over $g\mathrm{\ne }1$, and set $s(1)=1$. This gives a normalized section. Map an extension class to the cohomology class of the factor set of any normalized section. This is well defined by [L1].

Every class $[f]\in H^2(G,M)$ is hit: choose a normalized cocycle representative $f$, build the twisted product $M{\times }_{f}G$, and use its standard section $g\mapsto (0,g)$. The factor set of that section is exactly $f$, so step 1.1 sends the resulting extension to $[f]$.

If an extension $1\to M\stackrel{i}{\to }E\stackrel{\pi }{\to }G\to 1$ has normalized section $s$ with factor set $f_s$, then $\mathrm{\Psi }:M{\times }_{f_s}G\to E,\mathrm{\Psi }(m,g)=i(m)s(g),$ is an extension equivalence. Its inverse sends $x\in E$ to $(i^{-1}(xs(\pi (x){)}^{-1}),\pi (x)),$ and the factor-set identity shows that $\mathrm{\Psi }$ respects multiplication.

If two extensions define the same cohomology class, then after choosing normalized sections their factor sets are cohomologous. By [L2], the corresponding twisted products are equivalent extensions. Composing those equivalences with the ones from step 2.2 shows that the original extensions are equivalent. Thus the map of step 1.1 is injective.

Steps 2.1 and 3.1 show that step 1.1 is a bijection.