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We finish the classification, begun in two earlier papers, of all simplefusion systems over finite nonabelian $p$-groups with an abelian subgroup ofindex $p$. In particular, this gives many new examples illustrating theenormous variety of exotic examples that can arise. In addition, we classifyall simple fusion systems over infinite nonabelian discrete $p$-toral groupswith an abelian subgroup of index $p$. In all of these cases (finite orinfinite), we reduce the problem to one of listing all $\mathbb{F}_pG$-modules(for $G$ finite) satisfying certain conditions: a problem which was solved inthe earlier paper by Craven, Oliver, and Semeraro using the classification offinite simple groups.

Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: III