On subgroups of the affine group of the Galois ring of even characteristic

There are four nonabelian 2-groups containing a cyclic subgroup of index two: dihedral group, modular maximal-cyclic group, quasidihedral group, generalized quaternion group. The first three groups are contained in the affine group of the residue ring $Aff (\mathbb{Z}_{2^n})$ and are generated by the additive group and one of the three involutions of the multiplicative group. The generalized quaternion group is not contained in the affine group. When these groups are used together, the question arises about the existence of a ring in whose affine group all four groups are embedded. A natural candidate for such a ring is the Galois ring, which generalizes the residue ring and the Galois fiels. In this work, all subgroups of the affine group of a Galois ring of order $2^{2n}$ and characteristic $2^n$ generated by an element of additive order $2^n$ and element of order $2$ from the multiplicative group are described. The list of such groups is limited to the groups mentioned above, except the generalized quaternion group, as well as the $ZD_{2^{n+2}}$ and $ZM_{2^{n+2}}$ groups. All isomorphic embeddings of the generalized quaternion group $Q_{2^{n+1}}$ into the affine group of the Galois ring $Aff \left( GR (2^{2n}, 2^n) \right)$ are also described.