Uniformly describe period-two skew morphisms on cyclic groups

Determine a uniform description of all skew morphisms of period 2 on cyclic groups, extending the existing results on period-two skew morphisms and square roots of automorphisms.

Background

The skew morphisms classified in the paper have skew-type 4 and necessarily have period 2. Their classification therefore provides only a specific family of period-two skew morphisms on cyclic groups.

Earlier work had constructed and partially studied period-two examples, but the paper leaves unresolved whether all such skew morphisms can be described within a single uniform framework.