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 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.Fix a prime p. A fusion system over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms first formulated by Puig [Pu], and modeled on the Sylow theorems for finite groups. The motivating example is the fusion system of a finite group G with S ∈ Syl p (G), whose morphisms are those homomorphisms between subgroups of S induced by conjugation in G.The general theme in this paper is to study fusion systems over finite p-groups S that contain an abelian subgroup A S such that A F and C S (A) = A. In such situations, we let Γ = Aut F (A) be its automizer, try to understand what restrictions the existence of such a fusion system imposes on the pair (A, O p ′ (Γ )), and also look for tools to describe all fusion systems that "realize" a given pair (A, O p ′ (Γ )) for A an abelian p-group and Γ ≤ Aut(A). This paper is centered around one family of examples: those where p = 3, where O 3 ′ (Γ ) ∼ = 2M 12 , M 11 , or A 6 = O 3 ′ (M 10 ), and where A is elementary abelian of rank 6, 5, or 4, respectively. But we hope that the tools we use to handle these cases will also be useful in many other situations. Our main results can be summarized as follows:Theorem A. Let F be a saturated fusion system over a finite 3-group S with an elementary abelian subgroup A ≤ S such that C S (A) = A, and such that eitherAssume also that A F . Then A S, S splits over A, and O 3 ′ (F ) is simple and isomorphic to the 3-fusion system of Co 1 in case (i), to that of Suz, Ly, or Co 3 in case (ii), or to that of U 4 (3), U 6 (2), McL, or Co 2 in case (iii).Theorem A is proven below as Theorem 3.29 (case (i)) and Theorem 4.31 (cases (ii) and (iii)). As one consequence of these results, we give new 3-local characterizations of the three Conway groups as well as of McL and U 6 (2) (Theorems 5.1, 5.2, and 5.3).Cases (ii) and (iii) of Theorem A have already been shown in earlier papers using very different methods. In [BFM], Baccanelli, Franchi, and Mainardis listed all saturated fusion systems F with O 3 (F ) = 1 over a Sylow 3-subgroup of the split extension E 81 ⋊ A 6 , and this includes the four systems that appear in case (iii) of the above theorem. In [PSm],