In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps from the edge groups to the vertex groups are not necessarily injective. I will call these n.i. graphs of groups. There is already a question and some helpful answers about such n.i. graphs of groups here.

As noted in that linked question, given a connected n.i. graph of groups $\mathcal{G}$ we can go ahead and define the fundamental group $\pi_1(\mathcal{G})$ just as in the case of a normal graph of groups (either combinatorially as Serre does in "Trees" or topologically by constructing the appropriate graph of spaces and taking the fundamental group of its total space as in "Topological methods in group theory" Scott and Wall).

Is there a fintely presented group $G$ such that $G = \pi_1(\mathcal{G})$ for some n.i. graph of groups $\mathcal{G}$ with all of the vertex and edge groups of $\mathcal{G}$ finite, but $G$ can not be decomposed as the fundamental group of a (good ol' classic edge-map injective) graph of groups where all of the vertex and edge groups are finite?

What about the same question but where we replace "finite" with "finitely generated free" or "finitely generated abelian" or "finitely generated free abelian"??