On ARX-like ciphers based on different codings of $2$-groups with a cyclic subgroup of index $2$

A large number of block ciphers are based on easily and efficiently implemented group operations on $2$-groups such as the additive group of the residue ring $\mathbb{Z}_{2^m}$, the additive group of the vector space $V_{m}(2)$ over $\mathrm{GF(2)}$ and their combination. ARX-like ciphers use the operations of cyclic shifts and additions in $\mathbb{Z}_{2^m}$, $V_{m}(2)$. For developing techniques of building and analysing new symmetric-key block ciphers, we study group properties of $m$-bit ARX-like ciphers based on regular groups generated by $(0,1,\ldots,2^m-1)$ and different codings of permutation representations of nonabelian $2$-groups with a cyclic subgroup of index $2$. There are exactly four isomorphism classes of the nonabelian $2$-groups such as the dihedral group $D_{2^m}$, the generalized quaternion group $Q_{2^m}$, the quasidihedral group $SD_{2^m}$ and the modular maximal-cyclic group $M_{2^m}$. For such groups, we get imprimitivity criterions and give conditions on codings in order that the group of the ARX-like cipher should be equal to the symmetric group $S_{2^m}$. We also provide examples of three natural codings and their group properties.