  In this paper we find the fusion rules of the free wreath products
$\widehat{\Gamma}\wr_*S_N^+$ for any (discrete) group $\Gamma$. To do this we
describe the spaces of intertwiners between basic corepresentations which
allows us to identify the irreducible corepresentations. We then apply the
knowledge of the fusion rules to prove, in most cases, several operator
algebraic properties of the associated reduced $C^*$-algebras such as
simplicity and uniqueness of the trace. We also prove that the associated von
Neumann algebra is a full type $II_1$-factor and that the dual of
$\widehat{\Gamma}\wr_*S_N^+$ has the Haagerup approximation property for all
finite groups $\Gamma$.
