From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR APPLICATIONS, p. 18, I find:

Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above theorem?

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

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