This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into two:
1) Are there any favoured examples that you would recommend a look at? (Recommended references would be just as welcome.)
And the main question:
2) What sort of functors on the category of groups leave the rational cohomology unchanged? In particular is there a projection onto a special subcategory of groups that is in some way the right category to study?
I have a feeling that someone with a good knowledge of rational homotopy theory would be able to answer this question with relative ease.