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We consider orbit configuration spaces associated to finite groups actingfreely by orientation preserving homeomorphisms on the $2$-sphere minus afinite number of points. Such action is equivalent to a homography action of afinite subgroup $G\subset \mathrm{PGL}(\mathbb{C}^2)$ on the complex projectiveline $\mathbb{P}^1$ minus a finite set $Z$ stable under $G$. We compute thecohomology 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 thework 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 homotopytheory. We also prove the existence of an LCS formula relating the Poincar\'eseries of $C_n^G(\mathbb{P}^1\setminus Z)$ to the ranks of quotients ofsuccessive terms of the lower central series of the fundamental group of$C_n^G(\mathbb{P}^1 \setminus Z)$. The successive quotients correspond tohomogenous elements of graded Lie algebras introduced by the author in anearlier work. Such formula is also known for classical configuration spaces of$\mathbb{C}$, where fundamental groups are Artin braid groups and the rankscorrespond to dimensions of homogenous elements of the Kohno-Drinfeld Liealgebras.

Algebraic invariants of orbit configuration spaces in genus zero associated to finite groups