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Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,\psi)$ bethe 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 intertwinersand find its fusion semiring. Then, we prove some stability properties of thefree wreath product operation. In particular, we find under which conditionstwo free wreath products are monoidally equivalent or have isomorphic fusionsemirings. We also establish some analytic and algebraic properties of$\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. As a last result, we prove thatthe free wreath product of two quantum automorphism groups can be seen as thequotient of a suitable quantum automorphism group.

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