This is directly related to an initial question I had here. I want to followup this question with another one. Supposing I know $G\le S_n$ is a permutation group and that $C_1$ and $C_2$ are conjugacy classes of $G$ (same size and having permutations of same cycle types). Now I use GAP like in the previous post to determine if there is an $G$-automorphism mapping $C_1$ to $C_2$ but can I also know (by using GAP again?) if this automorphism is (naturally) induced by the (symmetric) normalizer $N(G)$ of $G$ in $S_n$. By "naturally induced" I mean the image of the map from the N/C Lemma $N(G) \rightarrow \mathrm{Aut}(G)$ (we may assume that the symmetric centralizer of $G$ is $1$).
Edit: I looked at GAP manual and I think that IsConjugatorAutomorphism should do the trick, no?
As you surmise, indeed
IsConjugatorAutomorphismtests exactly what you want -- it checks whether there is an element in the symmetric group on the points moved by $G$ that will induce the automorphism.Incidentally, if $G$ is transitive, a necessary and sufficient criterion for this (which should be in textbooks, bu I've not found it there) is that a point stabilizer is mapped by the automorphism to another point stabilizer.