Let $\pi: G\to G_1$ be a surjective group homomorphism to a finite group $G_1$ and let $S_1$ be a (finite) generating set of $G_1$. Assume $\mathrm{Cay}(G_1,S_1)$ is the Cayley graph of $G_1$ with respect to the generating set $S_1$. Define $S:=\pi^{-1}(S_1)$ and consider the Cayley graph $\mathrm{Cay}(G,S)$.

I would like to know if there is any criteria that allows one to construct an embedding of $\mathrm{Cay}(G_1,S_1)$ into $\mathrm{Cay}(G,S)$. Surly this is not always possible but I am looking for an example that existence of such embedding is possible. For example when $\pi: \mathrm{GL}_2(\mathbb{Z}_p)\to \mathrm{GL}_2(\mathbb{F}_p)$ is the reduction map and one can use for instance Hensel lemma to construct fibers.