Examples of non-free group actions on trees with finite edge-stabilizers

Question

I am interested in finding examples of finitely-generated non-free groups $H$ such that $H$ is a finite index subgroup of some group $G$ and $H$ acts without edge-inversion on some tree $T$ with finite edge-stabilizers. I am unsure what sort of constraints can be placed on $H$ in this context. I am familiar with Bass-Serre Theory but I am still new to it.

I would also be interested in the case that $H$ is free but its action on $T$ is not.

Any ideas/advice would be greatly appreciated.

Answer 1

As I mentioned in the comments, free products give you actions on trees with the properties you desire.

1. If $A$ or $B$ are non-free groups then the free product $A*B$ acts on its Bass-Serre tree with trivial (hence finite) edge stabilisers.
2. If $A$ and $B$ are free groups then the free product $A*B$ is free but acts non-freely on its Bass-Serre tree (as vertex stabilisers are non-trivial).

I wrote out a description of the Bass-Serre tree for a free product $A*B$ and the associated action here.

More exotic examples can be cooked up using HNN-extensions or free products with amalgamation (e.g. finite but non-trivial edge stabilisers in (1)).