Here is a statement having a proof that involved the CFSG.

Let $p$ be a prime, and $S$ be a nonabelian finite simple group such that $S$ is isomorphic to a subgroup of $S\_p$ with $p\mid |S|$. Then $\mathrm{Out}(S)$ is a $p'$-group (i.e. $p\nmid |\mathrm{Out}(S)|$).

**Question** Is there a CFSG-free proof for this statement?