On stable twist 2-groups of quasi-Hopf algebras

For any finite-dimensional quasi-Hopf algebra A, we define the notion of a stable twist of A. We show that stable twists form a strict 2-group $\2ust(A)$, which is equivalent to (A)^. Given a group homomorphism ρ: G → (A), we determine the obstruction in H³(G,ZG(A)) to lift $ρ$ to a G-action on A. We also establish a bijection between such liftings and cocentral cleft extensions of [G] by A. The main application of $\2ust(A)$ is the computation of (A), together with an explicit description of the corresponding tensor autoequivalences. We apply this framework to pointed Hopf algebras A that arise as central extensions of Taft algebras by cyclic groups, and we explicitly determine all of their tensor autoequivalences. The equivariantizations of A by (A) produce a new family of Hopf algebras (n,ξ) of dimension $2nm$, where $ξ$ is a scalar of order m n such that neither (n,ξ) nor its dual has the Chevalley property. In dimension 24, we prove that (6,-1) and its dual are precisely the Hopf algebras H for which neither H nor H^* has the Chevalley property. This completes the classification of nonsemisimple Hopf algebras of dimension 24.