If you parametrize your torsor by a cocycle in $H^1(-,G)$, the automorphism group is the twisted form of $G$ given by the same cocycle, where $G$ acts on itself by conjugations (while the torsor itself is the twisted form of $G$ acting on itself by shifts).

In terms of the article from the comment below, $Aut(P)=G\wedge^G P$, where $G$ acts on $G$ by inner conjugation (so it is an inner form of $G$ as in Remark 1.8)