On a class of power piecewise affine permutations on nonabelian groups of order $2^m$ with cyclic subgroups of index $2$

It is known that four nonabelian groups of order $2^m$, where $m \ge 4$, have cyclic subgroups of index $2$. Examples are well-known dihedral groups and generalized quaternion groups. Any nonabelian group $G$ of order $2^m$ with cyclic subgroups of index $2$ can be considered similar to the additive abelian group of the residue ring $\mathbb{Z}_{2^m}$, which is used as a key-addition group of ciphers. In this paper, we define two classes of transformations on $G$, which are called power piecewise affine. For each class we prove a bijection criterion. Using these criteria, we can fully classify orthomorphisms or their variations among described classes of power piecewise affine permutations.