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We compare the algebras of the quantum automorphism group offinite-dimensional C$^\ast$-algebra $B$, which includes the quantum permutationgroup $S_N^+$, where $N = \dim B$. We show that matrix amplification andcrossed products by trace-preserving actions by a finite Abelian group $\Gamma$lead to isomorphic $\ast$-algebras. This allows us to transfer variousproperties such as inner unitarity, Connes embeddability, and strong$1$-boundedness between the various algebras associated with these quantumgroups.

Crossed Product Equivalence of Quantum Automorphism Groups