Suppose $G$ is a finite group, $S \subset G$, such that $G = \langle S \rangle$, lets define the unlabelled Cayley graph of $G$ in respect to $S$ as $Cay_u(G, S):= \Gamma(V, E)$, where $V = G$ and $E = \{(a, b) \in G| ab^{-1} \in (S \cup S^{-1})\setminus\{e\}\}$.
Now, suppose $\phi:G \to H$ is a surjective homomorphism. Then it is quite obvious, that $\langle \phi(S) \rangle = H$.
My question is:
Is $Cay_u(H, \phi(S))$ always a minor of $Cay_u(G, S)$?
In the particular case, when $Ker(\phi) = \langle \langle A \rangle \rangle$ for some $A \subset S$, it is clearly true. But what's about the general case?