Let $X$ be a projective group scheme, over some base $S$, and let $G$ be a finite group acting on $X$ by $S$-isomorphisms.
I would like to understand if/when the quotient $X/G$ is representable by a Projective $S$-scheme.
When $X$ is affine, say $\text{Spec}A$ for some ring $A$, then it is possible to take $X/G$ to be the spectrum of the ring of invariants.
I understand that more generally, one would need to find a cover of my scheme by special affine pieces which are $G$-stable, and then glue them together.
It is unclear to me why it is indeed the case that it should always be possible to find such cover, and even then, why would the cover necessarily be representable by a projective scheme?