Garlands containing the general linear groups over a intermediate field

Let $K/k$ be a finite extension of fields with intermediate subfield $L$ and let $H=$GL$_L(k)$ be the general linear group of the all $L$-linear invertible mappings of the vector space of the field $k$ over $L$. It is proved that the intermediate between GL$_k(K)^H$ and the normalizae of $H$ subgroups form a goarland.