Equivariant Cohomology

I have a space X with an action of a group G. There is the cohomology of X and the group cohomology of G. I can do something that will degenerate to these two special cases when X or G are trivial.

First take the space   it is homotopic to X because EG is contractible. There is a G action on it, act on each factor. This action is free because it is free on EG. That means you can take a well defined quotient space which will not have any weirdness. Take the ordinary cohomology of that thing.

If G is trivial then EG is too and our quotient space just spits back X.
If G acts freely, then the constructed space will be homotopic to the plain old orbit space.
If X is a point, then the constructed space will be BG, and you remember what the cohomology of that is don't you?