  We consider orbit configuration spaces associated to finite groups acting
freely by orientation preserving homeomorphisms on the $2$-sphere minus a
finite number of points. Such action is equivalent to a homography action of a
finite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projective
line $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute the
cohomology ring and the Poincar\'e series of the orbit configuration space
$C_n^G(\mathbb{P}^1 \setminus Z)$. This can be seen as a generalization of the
work of Arnold for the classical configuration space $C_n(\mathbb{C})$
($(G,Z)=(\{1\},\infty$)). It follows from the work that
$C_n^G(\mathbb{P}^1\setminus Z)$ is formal in the sense of rational homotopy
theory. We also prove the existence of an LCS formula relating the Poincar\'e
series of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients of
successive terms of the lower central series of the fundamental group of
$C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond to
homogenous elements of graded Lie algebras introduced by the author in an
earlier work. Such formula is also known for classical configuration spaces of
$\mathbb{C}$, where fundamental groups are Artin braid groups and the ranks
correspond to dimensions of homogenous elements of the Kohno-Drinfeld Lie
algebras.
