Inductive Galois Alperin weight conjecture

Fix a prime $p$, a sufficiently large $p$-modular system $(\mathcal{K},\mathcal{O},F)$ with finite residue field, and a finite abelian group $\mathcal{H}\leq \operatorname{Aut}(\mathcal{K})$ preserving $\mathcal{O}$ and surjecting onto $\operatorname{Aut}(F)$. Let $G\unlhd A$ be finite groups, let $\operatorname{IBr}(G)$ be the irreducible Brauer characters of $G$, and let $\mathcal{W}(G)/G$ be the set of $G$-conjugacy classes of $p$-weights. For a character or weight, use the stabilizer notation appearing in the source, and let $\geqslant_c$ denote the central-isomorphism relation on $\mathcal{H}$-triples. Inductive Galois Alperin weight conjecture. There exists an $\mathcal{H}\times A$-equivariant bijection $\Omega:\operatorname{IBr}(G)\rightarrow\mathcal{W}(G)/G$ such that

$$(A_{\theta^{\mathcal{H}}},G,\theta)_{\mathcal{H}}\geqslant_c(\operatorname{N}_A(Q)_{\varphi^{\mathcal{H}}},\operatorname{N}_G(Q),\varphi)_{\mathcal{H}}$$

for any $\theta\in\operatorname{IBr}(G)$ and $(Q,\varphi)\in\Omega(\theta)$. This is the strengthened inductive form of the Galois Alperin weight conjecture; the source states that the inductive condition is intended to reduce to quasi-simple groups, but gives no resolution status.