I feel sort of silly asking this question. Unless I'm very much mistaken the paper I'm reading assumes the following statement:

Let $G$ be a finite group. We may embed it via the Cayley embedding into an ambient permutation group $G \leq S_{|G|}$. Then any automorphism of $G$ comes from conjugation by an element in $N_{S_{|G|}}(G)$.

Is this statement true?