The Meta-Isomorphism Conjecture with coefficients and finite wreath products

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let ${\mathcal C}$ be a class of groups closed under isomorphisms, subgroups, and quotients, and define ${\mathcal C}(G)=\{K\subseteq G\mid K\in{\mathcal C}\}$. Meta-Isomorphism Conjecture with coefficients and finite wreath products. For every finite group $F$, the wreath product $G\wr F=(\prod_FG)\rtimes F$ satisfies the coefficient Meta-Isomorphism Conjecture for ${\mathbf S}$ with respect to the family ${\mathcal C}(G\wr F)$. The statement imposes the conjectural assembly property on all finite wreath products; the supplied text gives no resolution status.