Commutator subgroup of a subgroup

Question

Suppose $H \subset G$ is a subgroup of finite index (assume normal if necessary). Must it be the case that $[H, H] \subset [G,G]$ is of finite index? (i.e. the map $H^{ab} \rightarrow G^{ab}$ is of finite kernel)

Answer 1

Let's take $G = D_{\infty} = \langle r, s : s^2 = 1, srs = r^{-1}\rangle$ and its subgroup $H = \langle r \rangle$. This must be a counterexample if I didn't make a mistake again.