TITLE: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
QUESTION [20 upvotes]: Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\oplus_{i \geq 1} G_{i}/G_{i+1}$ where $G_1 = G$, $G_i = (G_{i-1}, G)$, and the Lie bracket is induced by the commutator $(a,b)$ on $G$. Tensor $L$ with $_{\mathbb Z}k$ to get a Lie algebra $\hat L$ over $k$. 
Is there any relationship between the cohomology algebra of $G$, $H^{*}(G, k)$, and the cohomology of the (graded) Lie algebra $\hat L$, $H^{*}(\hat L, k)$? 
I realize I am not being precise about what category these cohomologies are being computed in, as in graded versus ungraded. I am hoping someone can tell me there is a relationship or that there is no connection. Perhaps there are conditions on $G$ that ensure these cohomology algebras are isomorphic? References?

REPLY [3 votes]: I just happened to stumble upon this old question.  The checked answer
refers to a 1999 paper for a spectral sequence that was studied in a 
1994 paper that addresses the question at hand:
Annetta Bajer. The May spectral sequence for a finite $p$-group stops.
J. Algebra 167 (1994), no. 2, 448–459.  
Here is the Math Review of that paper: ``Let $k$ be a field of characteristic $p>0$, let $G$ be a finite $p$-group, and let $gr kG$ be the graded $k$-algebra associated to the filtration of $kG$ by the powers of its augmentation ideal. The May spectral sequence of the title is a spectral sequence whose $E_2$-page is $Ext_{gr kG}(k,k)$ which converges to a filtration of $H^{*}(G,k)=Ext_{kG}(k,k)$. The author shows that this spectral sequence has only finitely many nonzero differentials, and deduces a similar result for a related spectral sequence converging to a filtration of $H^*(G,M)$ for $M$ a finitely generated $kG$-module.''  
The published reference for the MSS in general goes back to 1966: The cohomology of restricted Lie algebras and of Hopf algebras. 
J. Algebra (1966), 123--146. ([3] on my web page).