Exercise 10.4 in Silverman's Arithmetic of Elliptic curves states that two torsors $X/K$, $Y/K$ for the same elliptic curve $E/K$ are isomorphic as $K$-varieties if and only if the classes $[X]$, $[Y]$ in the Weil-Chatelet group are in the same $Aut(E)$-orbit.
Is there a reference for a more general version of this result, when $G$ is a group scheme over a base?
Thank you! Evgeny