# Residually finiteness for a factor group

Suppose we have a finitely presented residually finite group $G=\langle X\,; R \rangle$, two isomorphic finite subgroups $C$ and $D$ of $G$. The question is whether the group $H=\langle X\,; R, C=D \rangle$ is also residually finite. If not, can we prove a similar statement under some reasonable additional assumptions?

Here are some related comments. First, factor group of a residually finite group is not necessarily residually finite. So if the above question has an affirmative answer, the finite condition on $C$ and $D$ should be essential. Second, it may worth noting that, if the HNN extension $\langle X, t\,; R, C^t=D \rangle$ is considered instead of $H$, then it is again a residually finite group provided that $C$ and $D$ are finite groups.

• "$C=D$" means nothing. Maybe you want to fix an isomorphism $u:C\to D$ and consider the quotient $H$ of $G$ by the normal subgroup generated by $\{xu(x)^{-1}:x\in C\}$. – YCor Jan 4 '16 at 0:01