If $X$ is an infinite graph, $G$ is a group acting on $X$ with finite quotient; make $Y=X/G$ into graph of groups by attaching stabilizers at vertices and edges. Let $Z$ be a graph of groups, with graph equal to graph of $X/G$, all groups at vertices and edged being finite(but not all trivial), and suppose there is morphism of graph of groups $\phi:X/G\rightarrow Z$ , such that $\phi$ is graph isomorphism, and its restriction to vertex groups (and edge groups) is surjective homomorphism.

Does there exist a group whose action on $X$ gives the graph of groups $Z$.