Say I have an algebraic curve $C$ over a field $k$ and a group $G$ acting on $C$. Under what conditions on $C$ and/or the action of $G$ on $C$ can one conclude that $H^0(C,\Omega^1_C)^G = H^0(C/G,\Omega^1_{C/G})$ holds?
For example, if the morphism $C \to C/G$ has some ramification, what can one say? Could it be that as long as both $C$ and $C/G$ are smooth and projective, this equality holds, for example?