Nonrealizability of certain representations in fusion systems

For a finite abelian $p$-group $A$ and a subgroup $\mathrm{\Gamma }\le \text{Aut}(A)$, we say that the pair $(\mathrm{\Gamma },A)$ is fusion realizable if there is a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S\ge A$ such that $C_S(A)=A$, ${\text{Aut}}_{\mathcal{F}}(A)=\mathrm{\Gamma }$ as subgroups of $\text{Aut}(A)$, and $A$ is not normal in $\mathcal{F}$. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\mathrm{\Gamma }$ one of the Mathieu groups, that the only ${\mathbb{F}}_p\mathrm{\Gamma }$-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.