Lower central quotients can be extracted from group homology via spectral sequence built up from free simplicial resolution of a group. So, if your complex variety is aspherical, you probably know those Hodge structures because everything is as natural and functorial as it can be.

By a classical result of Magnus, for free groups we have isomorphism between free Lie ring on abelianisation $\mathcal LF_{ab} = \mathcal LH_1(F, \mathbb Z)$ and Magnus Lie ring $LG := \bigoplus L_nF = \bigoplus \gamma_i(F)/\gamma_{i+1}(F)$.
In general, this morphism is just epi. Now take free simplicial resolution $F_{\bullet} \twoheadrightarrow G$. We can look at exact couple defined by exact sequences $L_nF \to F/\gamma_{n+1}F \to F/\gamma_{n}F$ and associate with it graded Lie ring spectral sequence converging to $L_n(\pi_0 (F)) = LG$. First sheet is given by $E^1_{p, q} = \pi_q (L_p F)$, and $s$-th differentials have degree $(s, -1)$. Actually, $E^1$ differs from free graded Lie ring on group homology only by torsion (we can check it introducing analogous s. s. for augmentation powers filtration on $\mathbb ZF$ and looking at morphism between them induced by $G \hookrightarrow \mathbb Z G$; rationally it's split injection by PBW, and it's known that $LG \otimes Q \cong \Delta_{\mathbb Q}G$), zeroth row is always free Lie on $G_{ab}$, first column is the shifted by 1 integral homology of $G$, and stripes below $k$ depend only on $H_{\leq k + 1}$.

Also, we instantly prove Stallings' result about maps inducing isomorphism on factors by $\gamma_i$ (for $G \xrightarrow{f} G'$ $G$ is para-$G'$ iff $H_1(f)$ iso and $H_2(f)$ is epi) just by checking differentials degree. Expression for $\gamma_1/\gamma_2$ comes from boundary morphism in this s. s.

(I don't remember reference for that stuff, but it's not hard to convert those speculations to actual proof. See also Cochran&Harvey, http://arxiv.org/pdf/math/0407203.pdf and Ellis, "Magnus-Witt type isomorphism for non-free groups".)