First cohomology classifies complements up to kernel conjugacy

If $z:G\to A$ is a crossed homomorphism, [L2] makes ${\mathrm{\Gamma }}_z$ a complement to $A$ in $A⋊G$. By [L3], replacing $z$ by a cohomologous cocycle replaces ${\mathrm{\Gamma }}_z$ by an $A$-conjugate complement. Hence the rule $[z]\mapsto [{\mathrm{\Gamma }}_z]$ is well defined on $H^1(G,A)$.

Every complement $H\le A⋊G$ arises from some crossed homomorphism: the projection $H\to G$ is an isomorphism, so for each $g\in G$ there is a unique element of $H$ of the form $(z(g),g)$, and [L2] says the resulting map $z$ is a crossed homomorphism. Thus the map from step 1.1 is surjective on complement classes.

If ${\mathrm{\Gamma }}_z$ and ${\mathrm{\Gamma }}_w$ are $A$-conjugate, then [L3] says $w-z$ is principal, so $[z]=[w]$ in the quotient [L1]. Therefore the map of step 1.1 is injective.

Steps 1.1-2.2 give a bijection between $H^1(G,A)$ and the $A$-conjugacy classes of complements to $A$ in $A⋊G$.