From the lecture notes http://www.mimuw.edu.pl/~sjack/prosem/Cohen_Singapore.final.24.december.2008.pdf, p. 18, I find:
[![enter image description here][1]][1]

**Question 1:** Given a surface $S$, are there any methods to compute the fundamental group of $k$-th unordered configuration space
$$
\pi_1(F(S,k)/\Sigma_k)?
$$

**Question 2:** Given a group $G=\pi_1(F(S,k)/\Sigma_k)$, I find 
$
K(G,1)=BG.
$
Are there any methods to compute the cohomology ring (cup product structure)
$$
H^*(BG;\mathbb{Z}_2)?
$$
  [1]: https://i.sstatic.net/Qd9sP.png