The inductive Alperin weight condition for finite groups

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma$ be the Galois group of $k/\mathbb{F}_p$, and let $G$ be a finite group. Let $\operatorname{Aut}(G)$ act on the simple modules and weights, and for a central idempotent $b$ let $\operatorname{Aut}(G)_b$ denote its stabiliser. Inductive Alperin weight condition. (i) For any finite group $G$, there exists a bijection

$$\mathcal{S}(kG)\to\mathcal{W}(k\mathcal{O}_G)$$

commuting with the action of $\operatorname{Aut}(G)$. (ii) For any finite group $G$ and any central idempotent $b$ of $kG$, there exists a bijection

$$\mathcal{S}(kGb)\to\mathcal{W}(k\mathcal{O}_G,b)$$

commuting with the action of $\operatorname{Aut}(G)_b$. This condition is extracted from the inductive blockwise Alperin weight condition and is attributed in the paper to Navarro–Tiep and Späth.