Automorphism group of alternating group equals symmetric group

Statement

For $n\neq 2,3,6$, the automorphism group of the alternating group $\operatorname {Alt} (n)$, i.e., the alternating group of degree $n$, is isomorphic to the symmetric group of degree $n$, with the action given by conjugation in the symmetric group.

In other words, the automorphism tower of the alternating group of degree $n$ stabilizes after one step, at the symmetric group of degree $n$.

Related facts

• Automorphism group of finitary alternating group equals symmetric group
• Automorphism group of finitary symmetric group equals symmetric group
• Symmetric groups on finite sets are complete
• Symmetric groups on infinite sets are complete