Finite simple groups whose Sylow $2$-subgroups contain a cyclic subgroup of index $16$

In the paper the following main result is proved.
Theorem.Let $X$ be a finite simple group with Sylow $2$-subgroup $P$. Suppose that $P$ has a cyclic subgroup of index $16$. Then either the sectional $2$-rank of $X$ does not exceed $4$, or $|P|\leqslant2^8$, or $X\cong L_2(32)$.
A use of results of Gorenstein and Harada (RZh.Mat., 1975, 5A192), Kondrat'ev (RZh.Mat., 1977, 12A192), Beisiegel (RZh.Mat., 1977, 12A191) and Volker Stingl leads to the conclusion that finite simple groups whose $2$-subgroups have a cyclic subgroup of index $16$ are known.
Bibliography: 24 titles.