Scott Aaronson's blog, I find the description of string isomorphism as-
you’re given two strings $x$ and $y$ over some finite alphabet, as well as the generators of a group $G$ of permutations of the string indices. The problem is to determine whether you can transform $x$ to $y$ by applying a permutation in $G$. Or even more generally: given a string $x$, find a full set of generators for the subgroup of $G$ that fixes $x$.
Is $G$ an arbitrary permutation group i.e. in the definition of string isomorphism, can we condiser any permutation group $G$ ?
I have read another blog. Thanks.
It sure looks like $G$ can be any subgroup of $S_n$, where $n$ is the (common) length of $x$ and $y$. At least in the part you're quoting there's nothing that would restrict it to only some of these groups.