Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewitz Theorem implies that $H_1(X)$ is isomorphic to the abelianization of $\pi_1(X)$.
I'm thinking about invariants of 2-knots which can be extracted from the second homology of their complements, and I'm therefore very much interested in the answer to the following question:

What part of the fundamental group is detected by $H_2(X)$?

In particular, is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$?
Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?