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