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