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?](https://mathoverflow.net/questions/60794/relation-between-cohomology-of-ordered-and-unordered-configuration-spaces)



Relation between cohomology of ordered and unordered configuration spaces
The group $\Sigma\_k$ acts freely on $F(M,k)$, this implies that the orbits coincide with the homotopy orbits. In general, if a group $G$ acts on a topological space $X$ you have a spectral sequence


$$H\_\*(G,H\_\*(X,K))\implies H\_\*(X\_{hG},K)$$


where $K$ is any commutative field. The left hand side is the homology of the group $G$ with coefficients in the $G$ module $H\_\*(X,K)$. One way to construct this spectral sequence is to realize that you have a fiber sequence :
$$X\to X\_{hG}\to BG$$
in which the action of $G\simeq \Omega BG$ on the fiber $X$ is the action you started with. Then you can run a Leray-Serre spectral sequence.


Note that if $G$ is finite and $K$ has characteristic $0$, the left hand side collapses to $H\_\*(X,K)/G$ as mentioned in the question you are citing.


A possible reference for this is Hatcher's book project on spectral sequences <http://www.math.cornell.edu/~hatcher/SSAT/SSch3.pdf>


Your question is about cohomology rather than homology but since you are working over a field, assuming that $M$ has finitely generated homology, then the cohomology of $F(M,k)/\Sigma\_k$ in a given degree is the dual of the homology of $F(M,k)/\Sigma\_k$ in that degree.


