The conjugation representation of ${\mathrm{GL}}_2$ and ${\mathrm{SL}}_2$ over finite local rings

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G={\mathrm{SL}}_2(\mathbb{Z}\mathrm{/}{p}^r)$, where $r\ge 1$ and $p\ge 5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain–Tiep question in the affirmative and also prove analogous statements for ${\mathrm{SL}}_2$ and ${\mathrm{GL}}_2$ over any finite local principal ideal ring with residue field of odd characteristic.