I have been able to follow the hints given in Hungerford and I have proved the fact that if $\sigma \in{\rm Aut}(S_4)$ and $\sigma$ fixes all the sylow-3 subgroups of $S_4$ then it is an identity map, however how am I going to show that if $\phi:S_4\to{\rm Aut}(S_4)$ then the map is surjective?
Can someone explain with some examples as to why this will hold true (not any proof but examples as to why any other form of automorphsim except inner automorphism will not be an automorphism)