Enumeration of strong dichotomy patterns

We apply the version of Pólya-Redfield theory obtained by White to count patterns with a given automorphism group to the enumeration of strong dichotomy patterns, that is, we count bicolor patterns of $\mathbb{Z}_{2k}$ with respect to the action of $\operatorname{Aff}(\mathbb{Z}_{2k})$ and with trivial isotropy group. As a byproduct, a conjectural instance of phenomenon similar to cyclic sieving for special cases of these combinatorial objects is proposed.