Finitely presented subgroup of finitely presented group

Question

If I am given a group $G$, which is finitely presented by $\langle S \mid R \rangle$, and I am given a finitely presented subgroup $H$ of $G$. Is it true that $H$ takes the form $\langle T \mid R' \rangle$ for $T \subset S$ and $R' \subset R$?

Answer 1

No. This is not even true if $G$ is finite. Take the cyclic group $G:=\langle x \mid x^4=1 \rangle$, then $H:=\{ 1, x^2 \}$ is a finite subgroup (so clearly finitely presentable), but its only generating set is not s subset of the generators of $G$.