Let $G$ be a discrete group. For a $G$-CW complex $X$, let $H^G_{\bullet}(X)$ denote the Borel equivariant homology of $X$. There are also relative versions of this.

Here's my question. Let $X$ be a $G$-CW complex. The suspension $\Sigma X$ is then a $G$-CW complex in a natural way, and has two $G$-invariant base points coming from the two suspension points. Let $p_0$ be one of the suspension points. How can we compute $H^G_{\bullet}(\Sigma X,p_0)$ from $H^G_{\bullet}(X)$?

More generally, is there a good reference on Borel equivariant (co)homology where these sorts of foundational questions are worked out? All the references on equivariant homotopy theory I've looked at seem focused on different things, but Borel equivariant homology is what comes up in the things I'm working on right now.