  We give a description of all the cuspidal representations of
$\mathrm{GL}_4(\mathfrak{o}_2)$, where $\mathfrak{o}_2$ is a finite ring coming
from the ring of integers in a local field, modulo the square of its maximal
ideal $\mathfrak{p}$. This shows in particular the existence of representations
which are cuspidal, yet are not strongly cuspidal, that is, do not have orbit
with irreducible characteristic polynomial mod $\mathfrak{p}$. It has been
shown by Aubert, Onn, and Prasad that this phenomenon cannot occur for
$\mathrm{GL}_n$, when $n$ is prime.
