Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See The characters of binary modular congruence group, Bulletin of the American Mathematical Society. 79 (1973), no. 4.), P. Kutzko wrote the character tables of the groups $$G_{n}:=\text{SL}_{2}(\mathbb{Z}/p^{n+1}\mathbb{Z})/\pm I.$$ Let $C_{n} := \textbf{O}_{p}(G_n)$ - the $p$-core of $G_n$ -
Question: Does anyone know if the character tables of the groups $C_n$ have been written up, and if so where can I find them?
It might be possible that from knowing the character tables of the groups $G_{n}$ one can construct the character tables of the $C_n$'s (This doesn't sound right in general but the groups $(G_{n}, C_{n})$ have some nice properties). Here I list some properties of the groups $G_n$ and $C_n$ that I find interesting, and that I think could help in calculating the tables for $C_n$ having that one knows the ones for $G_n$.
The finite groups $G_n$ form a projective system (here $G_n \rightarrow G_m$ is the natural reduction map whenever $n \geq m$).
The $p$-core of $G_n$ is ``big'', meaning that for all $n$, and for all $P \in \text{Syl}_{p}(G_n)$ we have that $[P:C_n]=p$.
It is not hard to see that $C_n=Ker(G_n \rightarrow G_0)$. In fact, any normal subgroup of $G_n$ is the Kernel of some $G_n \rightarrow G_m$.
For all $n$ we have that $C_{n}/Z(C_{n}) \cong C_{n-1}$. Moreover, $Z(C_{n})$ is minimal normal subgroup of $G_n$ and $G_{n}/Z(C_{n}) \cong G_{n-1}$.
