Let $S$ be an infinite graph, $G$ is a group acting (effectively) on $S$ with finite quotient graph $S/G$. Make $S/G$ into graph of groups in obvious way by assigning stabilizers at vertices and edges.

Let $\tilde{S}$ be universal cover of $S$ and $H$ be a group acting (effectively) on $\tilde{S}$ with same quotient as graph $S/G$, but the graph of group may be different.

Question: Does there exist a subgroup $K$ of $G$ which acts on $S$, with quotient graph of groups $S/K$ and $\tilde{S}/H$ isomorphic?

[Here I am failing to use covering space theory directly, because here I am considering quotients spaces with some algebraic structures on them, namely graph of groups. This problem arise when I was studying Serre's "Trees".]