Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such that $G\hookrightarrow H$). Define $\mathcal{S}_{\mathrm{tf}}$ to be the class of finitely presented torsion-free groups which occur as the subgroup of some *torsion-free* hyperbolic group. Clearly $\mathcal{S}_{\mathrm{tf}}\subseteq \mathcal{S}$.

My question is: Are $\mathcal{S}_{\mathrm{tf}}$ and $\mathcal{S}$ equal?

That is, when studying the class of finitely presented torsion-free subgroups of hyperbolic groups $\mathcal{S}$, is anything lost by restricting to torsion-free hyperbolic groups?

there exist a hyperbolic (indeed virtually free) group $G$ with a torsion-free (normal) subgroup $N$ not contained in any torsion-free finite index subgroup of $G$. Namely, choose a $k$-generated infinite simple group with an element of order 2 (there are many); surject $G=F_k\ast C_2$ onto it with $C_2$ mapped injectively, and let $N$ be the kernel. Then $N$ is torsion-free and the only finite index subgroup of $G$ containing $N$ is $G$ itself, which is not torsion-free. $\endgroup$ – YCor Nov 8 at 16:14