The action of a finite group G on a subshift of finite type (SFT) X is called free if every point has trivial stabilizer and it is called inert if the induced action on the dimension group of X is trivial. We show that any two free inert actions of a finite group G on an SFT are conjugate by an automorphism of any sufficiently high power of the shift space. This partially answers a question posed by Fiebig. As a consequence, we obtain that every two free elements of the stabilized automorphism group of a full shift are conjugate in this group. In addition, we generalize a result of Boyle, Carlsen, and Eilers concerning the flow equivalence of G-SFTs.
