Let P denote the real projective plane. It has an action of the circle group S^{1}. (e.g. Let S^{1} act on the 2-sphere by rotations about an axis, then this action descends to the quotient P). I have a simple question.

What is the equivariant cohomology $H^\ast_{S^1}(P;\mathbb{F}_2)$?

Ideally I want a description as a module over $H^\ast_{S^1}(\mathrm{pt};\mathbb{F}_2)$, but I can't even manage to compute the structure as an abelian group myself.