Let $G$ be a group generated by some permutations $G = \langle A,B,C,D,E \rangle$. Suppose $G$ acts on a set $X$.
Given some subset $X' \subseteq X$ and an injective map $f : X' \to X$, How can I search for short words of the group whose action on $X$ agrees with $f$?
Basically I want to find short words like $ABC'ACD$ that permute a couple things to given locations, but I don't mind what it does to anything else.
So far I've looked at EpimorphismFromFreeGroup(G:names:=["A","B","C","D","E"]); and converted the generators of various stabilizers to words.