I have a morphism of schemes $X\to S$ which is very nice: flat, proper, finitely presented. I also have a finite group $G$ acting (faithfully, but not necessarily freely) on $X/S$.
1) I am quite sure that this does not suffice for the existence of a quotient $X/G$, but I am stuck on finding a counterexample.
Assume also that for every point of $S$ there is a Zariski open suset of $S$ containing $S$ and such that there exists a line bundle on $X$ which is ample on the inverse image. This should guarantee (am I right here?) tha there is a closed imbedding in $\text{Proj}(\mathcal{A})$ where $\mathcal{A}$ is a quasi-coherent algebra on $S$.
2) Does the quotient exist now?
(it surely exists locally on $S$).