Self-normalizing nilpotent subgroups of the full linear group over a finite field

It has been proved (Ref. Zh. Mat., 1977, 4A170) that in the full linear group $GL(n,q)$, $n=2,3$, over a finite field of $q$ elements, $q$ odd or $q=2$, the only self-normalizing nilpotent subgroups are the normalizers of Sylow 2-subgroups and that for even $q>2$ there are no such subgroups. In the present note it is deduced from results of D. A. Suprunenko and R. F. Apatenok (Ref. Zh. Mat., 1960, 13586; 1962, 9A150) that this is true for any $n$.