Abstract A cyclic complementary extension of a finite group 𝐴 is a finite group 𝐺 which contains 𝐴 and a cyclic subgroup 𝐶 such that A ∩ C = 1 G A C=\1₆\ and G = A C G=AC. For any fixed generator 𝑐 of the cyclic factor C = ⟨ c ⟩ C= c of order 𝑛 in a cyclic complementary extension G = A C G=AC, the equations c x = φ (x) c Π (x) cx= (x) c^ (x), x ∈ A x A, determine a permutation φ: A → A A A and a function Π: A → Z n A₍ on 𝐴 characterized by the following properties: φ (1 A)
Hu et al. (Tue,) studied this question.