Suppose I have a finitely presented group $G,$ and a subgroup $H$ of $G$ given by its finite generating set (given as words in the generators of $G.$ I want to know whether $H/[H, H]$ is finite. Is this question tractable (for your favorite definition of "tractable"  decidable would be a good start...)

It is undecidable even when $G$ is a direct product of two free groups. Look at Corollary C on page 2 in this paper. 

