Continuous linear extension operators on spaces of holomorphic functions on closed subgroups of a complex Lie group

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 ${\displaystyle {\pi }_1:V\to \Gamma }$ is finite and proper, then ${\displaystyle R_V:O(\Gamma \times G)\to Im{R}_V\subset O\left(V\right)}$ has a right inverse