Shapiro lemma for group homology

Apply the Axiom of Choice to the family of nonempty left cosets $G\mathrm{/}H$ to obtain a transversal $T$. The group-ring freeness supplier then gives $\mathbb{Z}[G]\cong {⨁}_{t\in T}t\mathbb{Z}[H]$ as right $\mathbb{Z}[H]$-modules. Restriction therefore sends free, hence projective, right $\mathbb{Z}[G]$-modules to projective right $\mathbb{Z}[H]$-modules. Exactness and the augmentation are unchanged on restriction, so the restricted $P$ is a projective right $\mathbb{Z}[H]$-resolution of the right trivial module.

Tensor associativity gives an isomorphism of chain complexes $P{\otimes }_{\mathbb{Z}[G]}(\mathbb{Z}[G]{\otimes }_{\mathbb{Z}[H]}M)\cong (\mathrm{Res}P){\otimes }_{\mathbb{Z}[H]}M$. The left and right sides compute respectively $H_n(G;{\mathrm{Ind}}_H^{G}M)$ and $H_n(H;M)$, so their homology gives the asserted natural isomorphism.