Let $S$ and $S'$ be subsets of size $k$ of $\mathfrak{S}_n$.

Are there any necessary or sufficient conditions to determine whether or not $S$ and $S'$ yield isomorphic Cayley graphs?

Assuming we are able to partition the $k$-subsets of $\mathfrak{S}_n$ into equivalence classes, where two sets are equivalent if they yield isomorphic Cayley graphs, is there an algorithm that visits each equivalence class exactly once, generating an arbitrary representative of each class as it goes?

To the best of my knowledge, answers to both questions are known only for transpositions: a set of transpositions generates $\mathfrak{S}_n$ iff its "transposition graph" (vertices = {1, 2, ..., n}, edges = transpositions) is connected; $S$ and $S'$ yield isomorphic Cayley graphs iff their transposition graphs are isomorphic; and therefore, an algorithm for listing all nonquivalent sets would be to generate all graphs on $n$ vertices and $k$ edges, labelling their vertices arbitrarily so as to yield a set of transpositions.