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 $\Gamma$?
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 AtkinLehner 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 $\mod p$. 


As noted by in the answer to this question: Automorphisms of $SL_n(\mathbb{Z})$ by @Guntram, Margulis super rigidity implies that an authomorphism of a lattice (which a finite index subgroup of $SL(n, \mathbb{Z})$ is) extends to an automorphism of $SL(n, \mathbb{Q}).$ So, the answer to your previous question Principal congruence subgroups of $SL(n, \mathbb{Z})$ works just as well here. 

