Find a non trivial homomorphism from Aut(S_4) to S_4

(Hint: consider the set of Sylow 3 -subgroups of S_4) , and deduce that Aut(S_4) is

isomorphic to S_4.

Actually we know that Inn(S_4) ~ S_4 , so S_4 < ~ Aut(S_4) .

It seems that first question is to prove the other direction ( Aut(S_4) < S_4)

and this can be achieved by the Cayley homomorphism.

We have 4 syllow 3 subgroups. Let Aut(S_4) act on S the set of sylow 3 subgroups.

so this induces f: Aut(S_4) ---> S_4 but we need to find f such that

f is non trivial and ker(f) = {0} (i.e. f is 1-1). Any ideas ??