Let $P$ and $X$ be algebraic varieties, $\pi:P\to X$ a morphism and $G$ an algebraic group acting on $P$. Serre calls the triple $(G,P,X)$ a fibre system an proves that if it is locally isotrivial (i.e. trivial in the etale topology) then the action of $G$ must be sharply transitive on each fibre of $\pi$.
Now my question is: Conversely if we assume that the the action of $G$ is sharply transitive on each fibre why it does not imply that $P$ is trivial in the Zariski topology? (It certainly cannot be true, but I don't see why).