Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C^∗-Algebras

We compare the algebras of the quantum automorphism group of finite-dimensional C-algebra, which includes the quantum permutation group, where. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group lead to isomorphic -algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong -boundedness between the various algebras associated with these quantum groups.