Exercise 13, § 6

Let $G$ be a group, $H$ a subgroup of $G$, $M$ a $\mathbf{Z}^{(H)}$-left module, $N$ a $\mathbf{Z}^{(H)}$-right module. Equip the group $\hat{M} = \mathrm{Hom}_{\mathbf{Z}(H)}(\mathbf{Z}^{(G)}, M)$ (resp. $\tilde{N} = N \otimes_{\mathbf{Z}(H)} \mathbf{Z}^{(G)}$) with the structure of left $\mathbf{Z}^{(G)}$-module (resp. right) deduced from the structure of right $\mathbf{Z}^{(G)}$-module of $\mathbf{Z}^{(G)}$.

Let $m : \hat{M} \to M$ and $n : N \to \tilde{N}$ be the $\mathbf{Z}^{(H)}$-homomorphisms such that $m(u) = u(1)$ for $u \in \hat{M}$ and

and let $i : H \to G$ be the canonical injection. Prove that the composed homomorphisms and are isomorphisms for all $q \geqslant 0$.