The free wreath product of a compact quantum group by a quantum automorphism group

Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,\psi)$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$. In this paper, we study the free wreath product $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. First of all, we describe its space of intertwiners and find its fusion semiring. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.