$TI$-subgroups in groups of characteristic 2 type

Properties of cyclic $TI$-subgroups of order 4 in finite groups are studied. A consequence of the results is the
Corollary. {\it Suppose that the $2$-group $A$ is a $TI$-subgroup of a finite group $G$, and that $F^*(G)$ is a simple group of characteristic $2$ type. Then either $A$ is elementary, or $F^*(G)\simeq G_2(3),$ $L_2(2^n\pm1),$ $L_3(3),$ $U_3(3),$ $U_4(3),$ $L_4(2),$ $U_4(2),$ $Sz(2^n),$ $U_3(2^n),$ $L_3(4),$ or $M_{11}$.}
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