For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?

Or if not, is it true when we replace $G$ by some subgroup? That is:

Let $H$ be a finite subgroup of an infinite group $G$. Is there always a subgroup $F \subseteq G$, such that the normal closure $\overline{H\cap F}^F$ of $H$ in $F$ has infinite index in $F$?

Edit:
As Derek points out, the Tarski monster is a counterexample. But the known Tarski monsters seem *not* to be *finitely presented*, see this answer: https://mathoverflow.net/a/26073/61732. So one could ask the above question for $G$ finitely presented. According to the last comment on the answer in the link, it is an open question if there are finitely presented Tarski monsters.