Cyclic complementary extensions of dihedral groups determined by automorphisms

A finite group G is called a cyclic complementary extension of a group A if A is embedded in G as a subgroup, along with a cyclic complementary subgroup C such that G = AC. It is well known that, by specifying a generator c of C, the commuting rule cx = φ(x)cΠ(x), x ∈ A, determines a permutation φ: A → A and a function Π: A → ℤ|c|, known as a skew morphism of A and an extended power power function of φ, such that φ(xy) = φ(x)φΠ(x)(y) for all x, y ∈ A. Conversely, recent work by Hu and Jajcay (2026) demonstrates that every cyclic complementary extension of a finite group A can be represented by a skew morphism φ of A and an extended power function Π: A → ℤn associated with φ, where n is a multiple of m = |φ|. Note that if Π(x) ≡ 1 (mod |φ|) for all x ∈ A,then φ is indeed an automorphism of A. Thus, skew morphisms encompass a broader scope, extending the concept of group automorphisms. In this paper, using the above correspondence, we determine the cyclic complementary extensions of the dihedral groups D2k determined by automorphisms of D2k.