Overgroups of order ${2^n}$ additive regular groups of a residue ring and of a vector space

The additive groups of the residue ring ${\mathbb{Z}_{{2^n}}}$ and of the vector space ${V_n}$ over the field $GF(2)$, as well as the group ${G_n}$ generated by these additive groups, share common imprimitivity systems and enter as subgroups into the Sylow 2-subgroup of the symmetric group $S({\mathbb{Z}_{{2^n}}})$. These groups are used in cryptography as an encryption tool with the operations of addition in ${V_n}$ and ${\mathbb{Z}_{{2^n}}}$. The permutation structure of the subgroups of the group ${G_n}$ is presented. The kernels of homomorphisms which correspond to various systems of imprimitivity, the normal subgroups, and some modular representations of the group ${G_n}$ over the field $GF(2)$ are described.