Let $G$ be a group with finite index subgroup $H$. Let $G^\prime = [G,G]$ denote the derived subgroup of $G$.

Is it true that $|G:H|<\infty$ implies that $|G^\prime: H^\prime|<\infty$.

If this is not true in general, is it true for a large class of groups? say, finitely generated.

Thanks to Mark Sapir for providing a simple counter-example to the general statement below.

What about if $G$ is nilpotent for example?