Non-inner non-simple fusion system for dihedral group:D8

This article describes a fusion system on dihedral group:D8 that is neither the inner fusion system (the one induced purely by inner automorphisms of the group) nor the simple fusion system. It is an intermediate fusion system]].

Strictly speaking, there are two such fusion systems if we fix the group beforehand concretely. However, the two fusion systems are isomorphic as fusion systems. More explicitly, they are interchanged by any outer automorphism of dihedral group:D8.

Explicit description
 This fusion system is realized, for instance, in the symmetric group of degree four. In fact, the symmetric group of degree four is in essence the only example -- any example admits this as a subquotient. In the table below, we describe the subgroups explicitly using this realization, i.e., in terms of D8 in S4.

Here is a description of the automorphism group from the fusion system in the two cases that it is not the full automorphism group:


 * For the group $$\{ e, x, a^2, a^2x \}$$, which is one of the Klein four-subgroups of dihedral group:D8, the automorphisms arising from the fusion system are the identity map and the automorphism that interchanges $$(1,3)$$ with $$(2,4)$$, while fixing $$$$ and $$(1,3)(2,4)$$.
 * For the whole group, the automorphisms arising from the fusion system are precisely the inner automorphisms. This is true for all fusion systems on dihedral group:D8 because the outer automorphism group of this group is a 2-group, so it has no 2'-automorphisms.

Realization in groups
Here are some examples of a group having dihedral group:D8 as a 2-Sylow subgroup such that this is the fusion system induced: