A. Vistoli, Z. Reichstein, Birational isomorphisms between twisted group actions, J. Lie Theory, Journal of Lie Theory 16 (2006), Number 4, 791--802.

Abstract: Let X be an algebraic variety with a generically free action of a connected algebraic group G. Given an automorphism f : G -> G, we will denote by X^f the same variety X with the G-action given by g : x -> f(g). x

V. L. Popov asked if X and X^f are always G-equivariantly birationally isomorphic. We construct examples to show that this is not the case in general. The problem of whether or not such examples can exist in the case where X is a vector space with a generically free linear action, remains open. On the other hand, we prove that X and X^f are always stably birationally isomorphic, i.e., X \times A^m and X^f x A^m are G-equivariantly birationally isomorphic for a suitable non-negative integer m.