I know there is a plethora of literature on how to construct quotients by groups, but my situation is quite particular, so I would appreciate if you could give me some hints or bibliographical references.
I'm interested in the following question: let $X$ be an algebraic variety defined over an embedded number field $k \hookrightarrow \mathbb{C}$. Assume that $X$ is smooth and projective and that it comes with the action of a finite group $G$. Then the quotient $X/G$ exists.
$\textbf{First}$: what is the best reference to learn the construction?
$\textbf{Second}$: what are the (scheme theoretic) properties of the "projection" $\pi: X \to Y$?
For instance, is it true that the direct image of the constant sheaf
$\pi_\ast \mathbb{C}_X$
on $X(\mathbb{C})$ is a local system on a certain open subset $U \subset Y$ excluding the singularities of $Y$?
Is it still true that the direct image by $\pi$ of a regular singular connection is still regular singular?
Thanks for your help !