As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $\ker f$ is finitely generated?



Residually finite group surjective to nonresidually finite group with finitely generated kernel
You can obtain such a surjection for any finitely presented non-residually finite group $H$ using Daniel Wise's residually finite Rips construction, which is the main result of this paper: [A Residually Finite Version of Rips's Construction.](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024609302001406)


