Let $G=G_1\times G_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the category of varieties and is a $G$-torsor locally trivial in the etale topology.
Is it true, that $X \rightarrow X/G_1$ exists in the category of varieties?
Is it an etale locally trivial $G_1$-torsor?
Is $X/G_1\rightarrow X/G$ an etale locally trivial $G_2$-torsor?
If not does it help to assume the $G_i$ to be reductive?
What if instead $G_1$ is only a normal subgoup of $G$ and $G_2$ the quotient group?