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