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

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi )$ and $\mathrm{\Gamma }$ a discrete group. We want to compute the fusion rules of $\widehat{\mathrm{\Gamma }}{\wr }_{\ast }\mathbb{G}$. First of all, we will revise the representation theory of $\mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $\psi$. We will also prove the simplicity of the reduced C*-algebra, when $\psi$ is a trace, as well as the Haagerup property of $L^{\mathrm{\infty }}(\widehat{\mathrm{\Gamma }}{\wr }_{\ast }\mathbb{G})$, when $\mathrm{\Gamma }$ is moreover finite.