gr.group theory - Does a Cayley graph on a minimal symmetric set of generators determine a finite group up to isomorphism?

Let $G = Z_4$ be the cyclic group on 4 elements, generated by $S = {-1,1 }$, let $H = Z_2 times Z_2$ be the Klein four group, generated by $T = {(0,1),(1,0)}$. Then $|S| = |T|$ and both Cayley graphs are isomorphic to $C_4$, the cycle of length 4.

For $n > 2$ each even cycle $C_{2n}$ is a Cayley graph for the cyclic group $Z_{2n}$ and for the dihedral group $D_n$ of order $2n$.

Another well-known example is the graph of a cube $Q_3$ which is a Cayley graph for the abelian group $Z_4 times Z_2$ and for the dihedral group $D_4$. In the previous example the dihedral group was generated by two involutions, while in the latter case it is generated by an involution and an element of order 4.

If only generators are counted, without their inverses, the first two examples do not give matching counts.