Is there a systematic way (e.g., an algorithm) to find all automorphisms of an arbitrary symmetric group $S_n$?
If not, can we simplify this question and ask again? That is, can there be some kind of properties which if n satisfies, then a systematic way to find all automorphisms of $S_n$ does exist?
(This question is asked as a generalization of one of my homework.)
Touching on what Lord Shark the Unknown said, $S_6$ is the only symmetric group with nontrivial outer automorphism group, which is isomorphic to $\Bbb{Z}/2\Bbb{Z}.$ The map which provides this is making the appropriate map that sends $$(ab)\mapsto (cd)(ef)(gh).$$ The map is entirely defined by how it is defined on these transpositions, since $S_n$ is generated by its transpositions.
See more about it here: https://groupprops.subwiki.org/wiki/Symmetric_groups_on_finite_sets_are_complete