Let $G=\mathbb Z_2$ and let $X \subset \mathbb P^5$ be a projective variety and the action of $G$ on $X$ is given by $1.(x_1,x_2,x_3,x_4,x_5,x_6)=-(x_6,x_5,x_3,x_4,x_2,x_1)$.
I need to compute the cohomology of the GIT quotient $X//G$. Now since the action is not free, I don't have a tool to compute the cohomology (I don't need equivariant cohomology).
So instead of computing the equivariant cohomology I want to compute the cohomology of the space $(X-X^G)//G$ and then go back to the space $X//G$. Note that the action of $G$ on $X-X^G$ is free and the cohomology can be computed easily.
For that I need to first understand the variety $(X-X^G)//G$ and relate it to the variety $X//G$. But I don't have any idea how to compute them. Any idea in this direction is highly appreciated. Thanks.