@Derek : Thanks for the correction! You explained it so well that it seems silly to edit the question to include this.

I tried. The Hopf formula seems to give me the answer for the quotients of the free group by the ordinary lower central series, but I couldn't make it work for the mod p version (for instance, the relations do not live in the commutator subgroup).

Edited it because I realized that I asked about general $H_k$ when I really only care about $H_2$ (as indicated in the title).