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?

Finitely presentable, non-Hopfian groups with Kazhdan’s property (T) and infinite outer automorphism group, Proc. Amer. Math. Soc. 135 (2007), no. 4, 951-959. – Andreas Thom Oct 22 at 11:27