A characterization of the groups $L_3(2^n)$

This note is concerned with finite groups in which a Sylow two-subgroup $S$ has an elementary Abelian subgroup $E$ of order $2^{2n}$, $n\ge2$, such that $E=A\times Z(S)$, $|A|=2^n$, and $C_S(a)=E$ for any involution $a\in A$.
It is proved that a simple group satisfying this condition is isomorphic to $L_3(2^n)$.