How can we show that every automorphism of $S_4$ is an inner automorphism ?
Note that:
Any automorphism preserves the conjugacy class of transpositions.( By counting size of conjugacy classes)
Any automorphism preserves whether two transpositions share an element (by looking at the order of their product)
Any automorphism permutes the four classes of transpositions {those that move 1}, {those that move 2}, {those that move 3}, {those that move 4}
This gives you a permutation on {1,2,3,4} and it is not hard to show that your automorphism is conjugation by this permutation