I came across a small question related to the Cayley graphs of semidirect products of the form $G=(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_q$.
Consider $Cay(G, S_1)$, where $S_1=\{a,b,c\}$, with $|a|=|b|=p, |c|=q$. When the vertices of the Cayley graph are arranged such that subgraphs corresponding to cosets of $(\mathbb{Z}_p \times \mathbb{Z}_p)$ can be clearly seen, I can observe $q$ cycles in the graph clearly too. I understood that between the subgraphs corresponding to the cosets of $(\mathbb{Z}_p \times \mathbb{Z}_p)$, the vertices are mapped based on a homomorphism $\theta: \mathbb{Z}_p \rightarrow Aut(\mathbb{Z}_p \times \mathbb{Z}_p)$ of the semidirect product and the edges are the edges due to the element of order $q$, which is $c$.
But, if we consider a $Cay(G, S_2)$, where $S_2=\{s,t\}$, with $|s|=|t|=q$, then the edges present are due to two elements of order $q$, and if we arrange the vertices of the graph so that the vertices corresponding to cosets of $(\mathbb{Z}_p \times \mathbb{Z}_p)$ can be identified as before, between each two subgraphs corresponding to cosets of $(\mathbb{Z}_p \times \mathbb{Z}_p)$, there are two types of edges connecting the vertices.
Then as in the earlier case can we identify the edges mapping between the subgraphs based on a homomorphism $\theta$? Are there two homomorphisms because of the presence of two elements of order $q$?
I am grateful if some one can help me in this regard.
Thank you very much in advance.