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We show that the restriction operator of the space of holomorphic functions on a complex Lie group to an analytic subset V has a continuous linear right inverse if it is surjective and if V is a finite branched cover over a connected closed subgroup Γ of G. Moreover, we show that if Γ and G are complex Lie groups and V ⊂ Γ × G is an analytic set such that the canonical projection ${\pi }_1:V\to \Gamma$ is finite and proper, then $R_V:O\left(\Gamma \times G\right)\to Im{R}_V\subset O\left(V\right)$ has a right inverse