Edit: the original question was imprecise, I'm sorry about that. I hope this is better.

Let $k$ be a field, $S/k$ any scheme, $G/k$ be an algebraic group and $X$ an $S$-torsor under $G$ (so $G$ acts simply transitively on the fibers of $G \to S$). Under what (sufficient/necessary/whatever) conditions on the schemes $X$, $G$, $S$, ... do we get an isomorphism $\overline{X} \cong \overline{G} \times_{\overline{k}} \overline{S}$, where the bar denotes base change to the algebraic closure of $\overline{k}$? For example, do we always have this isomorphism if $\text{Pic}\ \overline{S}$ is trivial?