Suppose $G$ is a finite linear group, and I have a $G$torsor $Z \to X$. Suppose also I have a morphism $f : Y \to X$ with some properties $P$. What should these properties $P$ be in order to make the fiber product $Y \times_X Z \to Y $ (with the natural projection) a $G$torsor over $Y$?
I think P should be empty. The group $G$ acts on the fibre product by acting on $Z$, pulling back a trivialising étale cover for $Z \to X$ gives a trivial étale cover for $Y \times_X Z \to Y$. No? 

