Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^*_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, contravariant functors $M:\mathcal{O}_G\to \mathcal{Ab}$ from the category of finite $G$-sets and $G$-maps to the category of abelian groups. This Bredon cohomology has now been generalized and extended in many ways by many authors. Although this question is manily about the "classical" setting described in Bredon's book, I would be interested also in answers which pertain to more sophisticated versions.

I've recently started trying to do some computations of Bredon cohomology, and have been able to compute the groups $H_G^i(X;M)$ in particular cases. Now I need to compute some cup products $$ H^i_G(X;M)\otimes H^{j}_G(X;N)\to H^{i+j}_G(X;M\otimes N), $$ and find I am lacking in tools, or guiding examples, which are available to me in the non-equivariant case. Therefore I ask:

What are some examples in the literature of calculations of cup products in the Bredon cohomology of specific $G$-spaces?

Perhaps one needs to resort to finding a $G$-equivariant approximation of the diagonal, or extending to an $RO(G)$-graded cohomology theory. Then I would appreciate references where this is done to produce calculations of products in the $\mathbb{Z}$-graded theory.

[1] *Glen E. Bredon*, MR 214062 **Equivariant cohomology theories**, *Lecture Notes in Mathematics, No. 34* .