  Let X be an algebraic variety with a generically free action of a connected
algebraic group G. Given an automorphism u of G, we will denote by X^u the same
variety X with the G-action given by twisted by u. V. L. Popov asked if X and
X^u are always G-equivariantly birationally isomorphic. We construct examples
to show that this is not the case in general, but we prove that X and X^u are
always stably birationally isomorphic, i.e., birationally isomorphic after
taking products with an affine space with trivial G-action.
