Let $k$ be an algebraically closed field, $X$ a quasiprojective $k$ scheme, $G$ a smooth linear algebraic group. We consider $f:P to X$ to be a principal $G$-bundle (here I’m assuming $f$ to be locally trivial in the etale topology). Given now another quasi projective $k$ scheme $F$ with a $G$ action, we consider the (right) $G$-action on $P times_k F$ defined by: $$g cdot (p,f)=(pg,g^{-1}f) .$$

I’ve seen claimed that the quotient $P times_k F/G$ always exists (with no condition on $G$), but I was not able to prove that. In particular, I was not able to specialize the standard topological proof to the algebrogeometric context, given that we have just etale local triviality.