The conjugation representation of the binary modular congruence group

Motivated by the study of an Hecke action on iterated Shimura integrals undertaken in [H], in this appendix to [H] we prove that, for any prime $p \geq 5$ and for any integer $n \geq 1$, every complex irreducible representation of $G=\mathrm{SL}_2(\mathbb{Z}/p^n \mathbb{Z})$ that are trivial on $\mathbf{Z}(G)$ appears as an irreducible constituent of the conjugation representation of $G$.