Arrangement of the subgroups that contain an unramified quadratic torus in the general linear group of degree 2 over a local number field ($p=2$)

Let $k$ be a dyadic local number field and let $K=k(\sqrt d)$ be an unramified quadratic extension. A complete description is suggested for the intermediate subgroups of the general linear group $\mathrm{G=GL}(2,k)$ of degree 2 over the field $k$ that contain the nonsplit maximal torus $T=T(d)$ (i.e., the image in $\mathrm G$ of the multiplicative group $K^*$ of the field $K$ under the regular embedding). In particular, the torus $T(d)$ is polynormal in $\mathrm{GL}(2,k)$. Bibliography: 11 titles.