Let $G_1$ be a circulant graph of prime order $p$. This implies that $G_1$ is the Cayley graph on $\mathbb{Z}_p$ with some generating set $S_1$. I am interested in knowing the characterizations of the generating sets $S$ of Cayley graphs $G$ on arbitrary $p$- groups with $S=S_1\cup S_2$, $S_2$ consisting of elements not in $\mathbb{Z}_p$, such that the vertex coloring of the graph $G_1$ can be extended to the vertices of $G$.

Specifically, I ask for what kind of and how many elements can the set $S_2$ have. Suppose I had just one element in $S_2$, then by taking cosets with respect to an element not in $S$ of the colored independent sets of $G_1$, we could easily produce a coloring with the same number of colors for the vertices of $G$. But, this process of taking cosets will not be fruitful if we have more number of elements. So how do we give a coloring extension in such a situation. What could be a possible limitation of the number of elements in $S_2$? Any hints? Thanks beforehand.