Infinite chains of successive normalizers in the general linear group

Let $K$ be a field of characteristics 0 or a field of characteristic 2 and of transcendence degree $\ge1$, and let $\mathrm{G=GL}(n,K)$ be the general linear group of degree $n\ge2$ over $K$. Further, let $1\le\rho\le\frac{n^2}4$. It is proved that in $\mathrm G$ there exist chains of subgroups $\{H_m\colon m\in\mathbb Z\}$, infinite in both directions, such that $H_m<H_{m-1}$, $H_{m-1}$ coincides with the normalizer $\mathcal N_\mathrm G(H_m)$, and every quotient group $H_{m-1}/H_m$ is an elementary Abelian group of type $(2,2,\dots,2)$ and of rank $\rho$. Bibliography: 7 titles.