I'm into some topics in group cohomology and I can't understand a basic thing about the induced actions on the cohomology modules.
Say $G$ is a group and $M$ is a $G$-module, namely a module with a $G$-action (formally speaking, it is a $\mathbf{Z}[G]$-module). Choose a normal subgroup $H\leq G$; then there is a restriction map $$H^1 (G,M)\longrightarrow H^1(H,M)$$ where the $H$-module structure over $M$ is obviously induced by the inclusion $H\subset G$.
I have been told, however, that the group $G/H$ acts on $H^1(H,M)$. I can't figure out what it is. Even if we look at $H^1(H,M)$ as the set of crossed maps modulo principal maps (i.e maps $f:H\longrightarrow M$ acting like $f(gh)=gf(h)+f(g)$ modulo those of the form $gm-m$ for some $m\in M$) I don't see an easy way to define the action.
What am I missing?
The action of $G/H$ is more evident from other interpretations of group cohomology, but in terms of crossed homomorphisms:
$G$ acts on the group of crossed homomorphisms $f:H\to M$ by $$(g\cdot f)(h)=gf(g^{-1}hg),$$ and if $f$ is principal then $g\cdot f$ is principal, so this induces an action of $G$ on $H^1(H,M)$.
If $g\in H$, then $$(g\cdot f)(h)=gf\left(g^{-1}(hg)\right) =gf(g^{-1})+f(hg)=gf(g^{-1})+f(h)+hf(g),$$ so $$(g\cdot f -f)(h)=gf(g^{-1})+hf(g),$$ which is principal, since $$0=f(1)=f(gg^{-1})=f(g)+gf(g^{-1}),$$ and so $$(g\cdot f-f)(h)=hf(g)-f(g).$$
So the action of $H$ on $H^1(H,M)$ is trivial, and so the action of $G$ factors through $G/H$.