The concept of a Cayley map is one of the central concepts of the part of Topological Graph Theory focused on highly symmetric maps. A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface which admits all the left multiplications by the elements of the group as automorphisms of the map. A Cayley map is regular if its automorphism group acts transitively on its set of darts. The automorphism group of a Cayley map is a cyclic complementary extension of its underlying group, where the cyclic part is generated by a special group mapping called skew-morphism. All cyclic complementary extensions are known to give rise to skew-morphisms, and all skew-morphisms give rise to specific cyclic complementary extensions called skew-products. However, not all cyclic complementary extensions are skew-products; leaving a gap in our understanding of cyclic complementary extensions of finite groups. Recently, together with Kan Hu, we have been able to fill this gap by introducing a generalization of the power function of a skew-morphism we call extended power function, and by finding a universal construction of cyclic complementary extensions of groups by skew-morphisms and their extended power functions. We have shown that all cyclic complementary extensions are constructed in this way.