Finite simple groups whose Sylow 2-subgroups possess an extra-special subgroup of index 2

Let $G$ be a finite simple group with Sylow 2-subgroup $T$. If there is an extra-special sub-group of index 2 in $T$, then $G$ is isomorphic to one of the following groups:
$$
\begin{array}{llllll} A_8,&A_9,&M_{11},&M_{12}\\ L_2(q),&L_3(q),&U_3(q),&G_2(q),&D_4^2(q),&PSp_4(q) \end{array}
$$
for an appropriate odd $q$.