I was reading Brown's Cohomology Theory of Finite groups and was wondering whether there's an example of the following.

Let $n\in \mathbb{N}$. Does there exist a $\mathbb{Z}_n$ module $M$ which is cohomologically trivial (i.e. $\smash{\hat{H}}^i(\mathbb{Z}_n,M)$ is trivial for each $i$) but such that there exists some $p,i\in \mathbb{N}$ such that $\smash{\hat{H}}^i(\mathbb{Z}_n,M/pM)$ is non-trivial?

In the case that $M$ has no $n$ torsion this is not possible by Lemma 8.6 of Brown's Cohomology of groups, I was wondering what goes wrong in the sense of a counterexample when $M$ has $n$ torsion.