For a given quasi-projective flat group scheme $G$ over a normal integral scheme with generic fiber $G_\eta$, we study models of \'etale torsors under $G_\eta$ with action of $G$, especially, their classification and quasi-sections. In the case where $G$ is the N\'eron model of an elliptic curve $G_\eta$ over an integral Dedekind scheme, we show that the action of $G_\eta$ on a torsor $X_\eta$ uniquely extends to an action of $G$ on the minimal regular model of $X_\eta$.