The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a question about $Conf(S^2, q)_{hSO(3)} = ESO(3) \times _{SO(3)} Conf(S^2, q)$, the $SO(3)$-equivariant homotopy type of this space.

Theorem: $Conf(S^2, q)_{hSO(3)}$ is aspherical for $q\geq 3$.

Proof (sketch): I will just write $C_q$ for $(Conf(S^2, q))_{hSO(3)}$. Projection to the first three coordinates $p:C_q \to C_3$ is a fibration (an equivariant version of the *Fadell-Neuwirth fibration*) with fiber $Conf(S^2-$ {3 points}, $q-3)$. But $C_3 \simeq *$, so the inclusion of the fiber $Conf(S^2-$ {3 points},$q-3)$ into $C_q$ is a homotopy equivalence. The fiber is a $K(\pi, 1)$ by an easy induction argument (using Fadell-Neuwirth again). QED.

Question: So $Conf(S^2, q)_{hSO(3)}$ is a $K(\pi, 1)$. What's $\pi?$