We know that principal congruence subgroups are characteristic in $SL(n,\mathbb Z)$. 
Suppose $\Gamma$ is a finite index subgroup of $SL(n,\mathbb Z)$ and $\Gamma\_m$ is a principal congruence subgroup of level m contained in $\Gamma$. Will it be characteristic in $\Gamma$?



Suppose $\Gamma_m$ is a principal congruence subgroup of level m contained in a finite index subgroup $\Gamma$ of $SL(n,\mathbb Z)$. Is $\Gamma_m$ characteristic in $\Gamma$?
This is false. Consider the case $n=2$, and let $p$ be a prime. Let $A=\left[\begin{array}{cc}0 & -1 \\\ p & 0\end{array}\right]
$ be an Atkin-Lehner involution (considered as an element of $PGL\_2(\mathbb{Q})$), and consider the subgroup
$\Gamma\_0(p) = \{ \left[\begin{array}{cc}a & b \\\ c & d\end{array}\right]\in SL\_2(\mathbb{Z}) | c\equiv 0(\mod p) \}$. Then one may check that for $B\in \Gamma\_0(p)$, $A^{-1} B A = \left[\begin{array}{cc}d & -c/p \\\ -pb & a\end{array}\right]
\in \Gamma\_0(p)$, so $A\in Aut(\Gamma\_0(p))$. Also, the principal congruence subgroup $\Gamma(p) \leq \Gamma\_0(p)$. However, consider the matrix $C=\left[\begin{array}{cc}1 & 0 \\\ p & 1\end{array}\right] \in \Gamma(p)$. Then one has $A^{-1} C A = \left[\begin{array}{cc}1 & -1 \\\ 0 & 1\end{array}\right] \notin \Gamma(p)$. Thus, the subgroup $\Gamma(p)\leq \Gamma\_0(p)$ is not characteristic. This extends to all $n$, taking the appropriate congruence subgroup by extending trivially $\mod p$. 


