On finite groups in which the generalized Fitting group of the centralizer of some involution is extraspecial (Q1239258)

In J. Thompson's \(N\)-group paper the following situation arises (section 13): for a maximal 2-local group \(M\), abelian normal subgroups of \(M\) have order at most 2. Hence \(Z\langle M\rangle = \langle z\rangle\) is of order 2, \(M=C_G(z)\) and \(F^*(M)\) is an extraspecial 2-group. In this paper this situation is considered in general and the following theorem is proved: Let \(G\) be a finite group and \(z\) an involution in \(G\) such that \(F^*(C_G(z))=Q\) is an extraspecial 2-group of width at least 2. Then one of the following holds: (1) \(z\in Z(G)\); (2) \(\langle z^G\rangle=F^*(G)\) is isomorphic to \(U_m(2)\), the \(m\)-dimensional unitary group over \(GF(2)\), or the second Conway group \(Co_2\); (3) \(z\not\in O_2(C_G(t))\) for some involution \(t\in Q\), in particular \(F^*(C_G(t))\) is not a 2-group, so \(G\) is not of characteristic 2 type; (4) \(z\) is fused in \(G\) to some noncentral involution of \(Q\).