Fusion systems related to polynomial representations of ${\mathrm{S}\mathrm{L}}_2(q)$

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of ${\mathrm{S}\mathrm{L}}_2(q)$, which includes the Sylow $p$-subgroups of ${\mathrm{G}\mathrm{L}}_3(q)$ and ${\mathrm{S}\mathrm{p}}_4(q)$ as special cases. The resulting list includes all Clelland–Parker fusion systems, a simple exotic fusion system discovered by Henke–Shpectorov, and a new infinite family of exotic examples.