Pushing up by \(2'\)-automorphisms of a Sylow \(2\)-subgroup (Q1801846)

The author continues to study the 2-local structure of finite simple groups of characteristic 2 type. Let \(G\) be a finite group such that a Sylow 2-subgroup \(S\) is contained in a unique maximal subgroup of \(G\) and \(C_ G(O_ 2(G)) \subseteq O_ 2(G)\). Let \(A\) be a group of automorphisms of \(S\) of odd order. Assuming that all simple sections of \(G\) are isomorphic to known simple groups, it is shown that some nonidentity \(A\)-invariant subgroup of \(S\) is normal in \(G\). This is a generalization both of a result of Stellmacher and of the author's previous work [(2.3) of Jap. J. Math., New Ser. 17, No. 2, 203-266 (1991; Zbl 0768.20007)].