Abstract Nonabelian 2-groups H containing a cyclic subgroup of index 2 are dihedral groups, generalized quaternion groups, quasidihedral groups and modular maximal-cyclic groups. Earlier the authors introduced the classes of piecewise quasiaffine transformations on an arbitrary nonabelian 2-group H with a cyclic subgroup of index 2. For the generalized group of quaternions of order 2 m we have obtained a complete classification of orthomorphisms, complete transformations and their left analogues in the class of piecewise quasiaffine transformations under consideration. This paper presents a similar classification for the remaining three groups (the dihedral group, the quasidihedral group and the modular maximal-cyclic group).
Pogorelov et al. (Mon,) studied this question.