Properties of permutation representations of nonabelian $2$-groups with a cyclic subgroup of index $2$

For all nonabelian $2$-groups with cyclic subgroup of index $2$ (the dihedral group $D_{2^m}$, the generalized quaternion group $Q_{2^m}$, the modular maximal-cyclic group $M_{2^m}$, the quasidigedral group $SD_{2^m}$) we describe properties of regular permutation representations. For each group we characterize all nontrivial imprimitivity systems and corresponding homomorphisms.