I believe the following statement is true:

Given a complex analytic map $f:\Delta\to V/G$, where $\Delta$ is a disc in $\mathbb{C}$, $V$ a finite dimensional complex vector space and $G$ a finite subgroup of $GL(V)$, then $f$ admits an analytic lift $\tilde f:\Delta'\to V$ up to a ramified cover. More precisely, there exists a ramified cover $r:\Delta'\to\Delta$ such that $f\circ r = \pi\circ \tilde f$ where $\pi:V\to V/G$ is the canonical projection.

I think I have a relatively elementary proof. However, this statement sounds very much like a "classical" result, but I have been unable to find a reference. Does anyone knows to whom I should attribute this result ?

Thanks a lot