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We study orbit configuration spaces $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$obtained from the action of a finite homography group $G$ on $\mathbb{P}^1$. Weconstruct a flat connection on the orbit space with values in a Lie algebra$\hat{\mathfrak{p}}_n(G) $. We establish an isomorphism of filtered Liealgebras between $\hat{\mathfrak{p}}_n(G)$, the Malcev Lie algebra of thefundamental group of $\mathrm{Cf}_G(n,\mathbb{P}^1_*)$ and the degreecompletion of the associated graded to the latter Lie algebra. Theseisomorphisms are obtained using the monodromy representation of the connectionand the study of the fundamental group.

Sur certains espaces de configurations associ\'es aux sous-groupes finis de $\mathrm{PSL}_2(\mathbb{C}) $