homotopy theory - Classifying Space of a Group Extension

Yes. The principal bundles are the same and your guess that BA is an abelian group is exactly right. A good reference for this story, and of Segal's result that David Roberts quotes, is Segal's paper:

G. Segal. Cohomology of topological groups, Symposia Mathematica IV (1970), 377- 387.

The functors E and B can be described in two steps. First you form a simplicial topological space, and then you realize this space. It is easy to see directly that EG is always a group and that there is an inclusion G --> EG, which induces the action. The quotient is BG. Under suitable conditions, for example if G is locally contractible (which includes the discrete case), the map EG --> BG will admit local sections and so EG will be a G-principal bundle over BG. This is proven in the appendix of Segal's paper, above. There are other conditions (well pointedness) which will do a similar thing.

The inclusion of G into EG is a normal subgroup precisely when G is abelian, and so in this case BG is again an abelian group.

I believe your question was implicitly in the discrete setting, but the non-discrete setting is relevant and is the subject of Segal's paper. Roughly here is the answer: Given an abelian (topological) group H, the BH-princical bundles over a space X are classified by the homotopy classes of maps [X, BBH]. When H is discrete, BBH = K(H,2). If X = K(G,1) for a discrete group G, these correspond to (central) group extensions:

H --> E --> G

If G has topology, then the group extensions can be more interesting. For example there can be non-trivial group extensions which are trivial as principal bundles. Easy example exist when H is a contractible group. However Segal developed a cohomology theory which classifies all these extensions. That is the subject of his paper.