The simplest overgroups of regular permutation representations of nonabelian $2$-groups with a cyclic subgroup of index $2$

For any nonabelian $2$-group $H_m$ with a subgroup of index $2$ (namely the dihedral group $D_{2^m}$, the generalized quaternion group $Q_{2^m}$, the modular maximal-cyclic group $M_{2^m}$, the quasidihedral group $SD_{2^m}$) we consider its simplest overgroups. In this way we describe properties of the group generated by the right and the left regular permutation representations of any $H_m$ including its structure, order, center, rang and estimate of the minimal degree. We characterise its automorphism group and all isomorphic embeddings of $H_m$ (of order $2^m$) into the affine group of the residue ring $\mathbb{Z}_{2^{m - 1}}$ if such embeddings exist.