Components and realizability of fusion systems

For $p=2$ it is known that a saturated fusion system is realizable if and only if each of its components is realizable by a finite quasisimple group. For odd primes $p$ this is false. Building on work of Broto, M{\o}ller, Oliver, and Ruiz, we show however that a fusion system $\mathcal{F}$ is realizable if and only if for each of its components $\mathcal{C}$ there exists a realizable subnormal subsystem $\mathcal{E}$ of $\mathcal{F}$ with $O^{p^\prime}(\mathcal{E})=\mathcal{C}$.