I have seen the following claim without proof in more than one paper, but it is sufficiently general that I suspect it is stated too strongly to be true:
Let $G$ be an affine group scheme (say, over a field of characteristic zero), let $X$ be a scheme (smooth over the same field), and let $P \to X$ be a $G_X$-torsor. If $Y$ is a scheme with a $G$-action, then the associated bundle $P \times^G Y$ is a scheme.
I can use fpqc descent to show that this is true when $Y$ is affine (or some other effective fpqc descent class), and I can use Zariski descent when $P$ is Zariski-locally trivial. In full generality, I suspect one can assemble known counterexamples of descent to falsify this claim, but I have been unable to do so.
Question: Is there a counterexample known? Failing that, is there a proof of the claim?
I'm somewhat more interested in the case where $G$ is connected, but here is a candidate that I don't know how to prove: Take $X$ to be a smooth curve with a nontrivial étale double cover $P$ (with $G$ constant of order 2), and set $Y$ to be Hironaka's 3-fold with involution (whose quotient sheaf is not a scheme).
