Restriction, inflation, and the quotient conjugation action on first cohomology

Let $N⊴G$ and let $A$ be an abelian $G$-group.

The restriction map

$\mathrm{Res}:H^1(G,A)\to H^1(N,A)$

is induced by restricting a crossed homomorphism $z:G\to A$ to $N$.

The inflation map

$\mathrm{Inf}:H^1(G\mathrm{/}N,A^N)\to H^1(G,A)$

is induced by pulling a cocycle $\bar{z}:G\mathrm{/}N\to A^N$ back along the quotient map:

$\mathrm{Inf}(\bar{z})(g)=\bar{z}(gN)\mathrm{.}$

For $g\in G$ and a crossed homomorphism $z:N\to A$, define another crossed homomorphism by

$(g\star z)(n)=g\cdot z(g^{-1}ng)\mathrm{.}$

Passing to cohomology classes, this action depends only on the coset $gN$, so it gives the quotient conjugation action of $G\mathrm{/}N$ on $H^1(N,A)$. If $M$ is any $G$-group, the same formula defines the quotient action on nonabelian $H_{\mathrm{n}\mathrm{a}\mathrm{b}}^1(N,M)$.