The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme $T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$ on $X(T)$ is simply transitive.
A pseudo $G$-torsor $X$ is trivial if and only if the morphism $X \to S$ has a section.
Proof. Omitted. $\square$
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