Let $A$ be an Abelian group, and let $G$ be a finite group which acts on a finite set $M$, such that a subgroup $H$ acts trivially on $M$ and $G/H$ acts freely on $M$. I can define a $G$-module $A^M$ (i.e. Abelian group with $G$-action), with the $G$-action inherited from $M$. By restricting onto $H$ I get a homomorphism from $H^3(BG,A^M)$ to $H^3(BH, A)$. My intuition is that it should be an isomorphism because only the action of $H$ on $A^M$ is "sufficiently localized" to correspond to any non-trivial data. Is this correct?



Group cohomology with coefficients in a permutation module
I'm not sure I understand your intuition, but the statement is true. It's a special case of [Shapiro's Lemma](https://en.wikipedia.org/wiki/Shapiro's_lemma): if $H$ is a subgroup of $G$, $A$ is a $\mathbb{Z}H$-module, and $A^G$ is the coinduced module, then
$$H^\ast(G,A^G)\cong H^\ast(H,A).$$


