Can someone point me to a good (hopefully simple and brief) place to read about the basics of homotopy orbits for pointed spaces?

More detail:

As I understand it, in the unpointed case,
we use the projection $\mathrm{pr}_X: EG \times X \to X$
to replace the $G$-shaped diagram $X$ with the cofibrant $G$-shaped
diagram $EG\times X$; and then the homotopy orbit space is the
homotopy colimit of the diagram $X$, which is the categorical
colimit of $EG\times X$.

I'd like a description of the pointed case in which it is shown that $EG_+ \wedge X\to X$ is a cofibrant replacement in the appropriate category of diagrams.

Even More (Inspired by T. Goodwillie's comment): This is what I think, too, but I feel like I'm missing something, because in the pointed case, why can't I just use the cone on $G$ instead of $EG$? It is contractible and $G$ acts freely on the nonbasepoint cells, right? So I could just as easily use $CG_+ \wedge X$. but this is clearly wrong!

EDIT: Cone idea is wrong because of the cone point, so I feel pretty good about my understanding of the theory. I'd still like a good reference.