Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group action of $\Sigma_k$, the symmetric group on $k$ letters.

Let $p$ be an odd prime. What is the relation between

$H^*(F(M,p);\mathbb{Z}/p\mathbb{Z})$ and $H^*(F(M,p)/\Sigma_p;\mathbb{Z}/p\mathbb{Z})$?

Suppose the cohomology algebra $H^*(F(M,p);\mathbb{Z}/p\mathbb{Z})$ is known. Is there any method, including spectral sequence, to get the cohomology algebra $H^*(F(M,p)/\Sigma_p;\mathbb{Z}/p\mathbb{Z})$? Could you give some references?

Thanks.

Note: when the coefficient field is of characteristic $0$ or of characteristic $q$, for $q$ not dividing $o(\Sigma_k)=k!$, the question is answered at Relation between cohomology of ordered and unordered configuration spaces?