Cuspidal representations which are not strongly cuspidal

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.