String Isomorphism

An important problem in computational complexity.

String

$ S $: a set.

$ A $: an alphabet. Yet another set.

$\sigma : S \rightarrow A$: a string. Map from $S$ to $A$.

$p:S \leftrightarrow S$: a bijection from $S$ to $S$.

${Sym}_S = \{ p \}$: group of all permutations of $S$. Symmetric group.

$G$: subgroup of ${Sym}_S$.

String Isomorphism

$\sigma_1$: string.

$\sigma_2$: string.

$G$: subgroup of ${Sym}_S$.

Question

Is there some $ p \in G $ such that $ p(\sigma_1) = \sigma_2 $ ?