Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- II

We classify all corefree fusion systems on a Sylow 3-subgroup of the sporadic groups ${\mathrm{F}\mathrm{i}}_{24}^{\mathrm{\prime }}$ and $\mathrm{M}$. The program to classify of all corefree fusion systems on Sylow $p$-subgroups of sporadic groups has been running for many years, comprising of work done in several papers by numerous authors. We complete this program for $p$ odd. In the process, we expand the theory of proto-essential subgroups and describe algorithms that make the process of finding all corefree fusion systems on a $p$-group better by several orders of magnitude.