Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one point?
I know that this cannot happen when $G$ is abelian, but what about the general case?
(EDIT: a branch point is the image in $C/G$ of a ramification point)