 Pentad construction of the exceptional alternating-group automorphism

The six pentads are the six one-factorizations of the complete graph on six points. Permuting the ground-set points permutes these factorizations. The action sends a transposition to three disjoint transpositions and a $3$-cycle to two disjoint $3$-cycles. Its restriction to $A_6$ is faithful by simplicity, and the full action of $S_6$ is faithful because an order-two kernel would be central. It is consequently an automorphism of $S_6$. Its restriction preserves $A_6$ and changes $3$-cycle type, so it cannot be induced on $A_6$ by any conjugation from $S_6$.