  We study a certain family of simple fusion systems over finite $3$-groups,
ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and
$A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to
the $3$-fusion systems of almost simple groups. As one consequence, we give new
$3$-local characterizations of Conway's sporadic simple groups.
