I think this is a pretty neat question. Wikipedia says that f([G,G]) is a subgroup of [H,H]. If we can show that
, then it should follow that f is surjective. But I don't see how p-groups are going to help ... ... ... is it possible to use the Sylow theorems?
owlpride, if you've already figured solved the problem, please post your solution. I'd enjoy seeing it