Centralizers of normal subgroups and the $Z^*$-Theorem

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$-subgroup $S$, the centre of $G/O_{p^\prime}(G)$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that seems a priori more general: For any normal subgroup $H$ of $G$ with $O_{p^\prime}(H)=1$, the centralizer $C_S(H)$ is expressed in terms of the fusion system $\mathcal{F}_S(H)$ and its normal subsystem induced by $H$.