An Endomorphism of a Finitely Generated Residually Finite Group

Let $\varphi :G\to G$ be an endomorphism of a finitely generated residually finite group. R.~Hirshon asked if there exists~$n$ such that the restriction of $\varphi$ to ${\varphi }^n(G)$ is injective. We give an example to show that this is not always the case.