The symmetric group of order $720=6!$, $S_6$, is the only symmetric group with an outer automorphism. This suggests the following:
Question: Is $A_6$ characteristic in $S_6$, where $A_6$ is the alternating group of order $360$?
Clearly, $A_6$ is normal in $S_6$, so if it is not characteristic, then some outer automorphism of $S_6$ must map some even permutation to an odd permutation.
If $n \ne 6$, then since all automorphisms of $S_n$ are inner, $A_n$ is characteristic in $S_n$.