The Fibered Meta-Isomorphism Conjecture for equivariant homology theories

Let $\Gamma$ be a group, let $(G,\psi)$ be a group over $\Gamma$ with homomorphism $\psi\colon G\to\Gamma$, let ${\mathcal H}^?_*$ be an equivariant homology theory over $\Gamma$, and let ${\mathcal F}$ be a family of subgroups of $G$. Fibered Meta-Isomorphism Conjecture. For every group homomorphism $\varphi\colon K\to G$, the group $K$ satisfies the Meta-Isomorphism Conjecture for the $K$-homology theory ${\mathcal H}^{K,\psi\circ\varphi}_*$ and the family $\varphi^*{\mathcal F}=\{H\subseteq K\mid\varphi(H)\in{\mathcal F}\}$. This fibered formulation requires the assembly conjecture after every change of group; the supplied text gives no resolution status.