Automorphism group of alternating group equals symmetric group

Statement
For $$n \ne 2,3,6$$, 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