If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth.

My question goes a little bit backward. Assume that $X$ is a variety (irreducible, reduced and separated scheme of finite typer over $\mathbb{C}$) with a faithful action of finite group $G$ on it.

Assume that the quotient $X/G$ is smooth. Can we say something on $X$ (is it Cohen-Macaulay?) Or on the quotient morphism $p : X \rightarrow X/G$ (is it flat?).

Thanks a lot!