Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ?

The first example for a group that does not have this property is a Baumslag-Solitar group $BS(1,m)= \langle a,b| bab^{-1}=a^m\rangle$. We have $a^{m^k}=b^kab^{-k}$ for each $k$. This shows for example that the inclusion of the subgroup generated by $a$ is not a quasiisometric embedding.

Then one can consider the class of groups with the property that the inclusion of any subgroup is a quasiisometric embedding. Has this class been studied?