I am sure the answer to this question is well-known, but 


It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a topological space $X$ elements of $H^2(X,\mathbb Z)$ are in natural bijection with complex line bundles on $X$.


My question is thus:


What is the direct correspondence between extensions of $G$ and line bundles on $BG$?


That is, given an explicit line bundle $L$ how does one construct an explicit group extension $E$ such that the two give the same cohomology class and vice versa?



Group Extensions and Line Bundles on $BG$
If $L \to BG$ is the complex line bundle, take the unit sphere bundle $S^1 \to S(L) \to BG$ and take $\pi\_1$.


