Let $G$ be a finite group with subgroup $H$. Suppose there is an unknown $\mathbb{Z}[G]$-module $M$, where we know $M_1 = \mathrm{res}^G_H M$ and $M_2 = M\otimes_{\mathbb{Z}[H]}\mathbb{Z}$. Is there some way of classifying the possibilities for $M$, in terms of $M_1$ and $M_2$ (maybe some cohomology group)?
(I'm actually interested in the case where $G$ is profinite and $\mathbb{Z}$ is replaced by $\mathbb{Z}_p$, but realized I didn't know the more basic version of this.)