  Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The
first author and Zalesskii have previously given a characterization of
permutation modules for $\mathbb{Z}_pG$ in terms of modules for $G/N$, but the
necessity of their conditions was not known. We apply a correspondence due to
Butler to demonstrate the necessity of the conditions, by exhibiting highly
non-trivial counterexamples to the claim that if both the $N$-invariants and
the $N$-coinvariants of a given lattice $U$ are permutation modules, then so is
$U$.
