  For a finite abelian $p$-group $A$ and a subgroup $\Gamma\le\text{Aut}(A)$,
we say that the pair $(\Gamma,A)$ is fusion realizable if there is a saturated
fusion system $\mathcal{F}$ over a finite $p$-group $S\ge A$ such that
$C_S(A)=A$, $\textrm{Aut}_{\mathcal{F}}(A)=\Gamma$ 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 $\Gamma$ one of the
Mathieu groups, that the only $\mathbb{F}_p\Gamma$-modules that are fusion
realizable (up to extensions by trivial modules) are the Todd modules and in
some cases their duals.

Group Theory