I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight.

Let $G$ be a group with an injective endomorphism $\phi$, then the HNN-extension $$G_\phi = \left<G,t \mid t^{-1} gt= \phi(g) \right> $$ is called the ascending HNN-extension of $G$ determined by $\phi$.

**Questions**:

Does there exist a finitely generated, torsion group $G$, with an injective but non-surjective endomorphism $\phi$, such that $G_\phi$ is residually finite?

**Thoughts**: Clearly, $G$ needs to be residually finite. In this paper Sapir & Wise showed that an ascending HNN extension of a finitely generated residually finite torsion group need not be residually finite. I was wondering, given the fact that $G_\phi$ is residually finite, can $G$ be a torsion group?

I tried to show that such a group does not exist. This paper contains the criterion for the HNN extension to be residually finite, but I can't seem to get a contradiction.

Any reference/examples would be really appreciated, thanks in advance.

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