Cyclic complementary extensions and skew-morphism

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>∩</m:mo> <m:mi>C</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mn>1</m:mn> <m:mi>G</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> A\cap C=\{1_{G}\} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mi>C</m:mi> </m:mrow> </m:mrow> </m:math> G=AC . For any fixed generator 𝑐 of the cyclic factor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:mi>c</m:mi> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> C=\langle c\rangle of order 𝑛 in a cyclic complementary extension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mi>C</m:mi> </m:mrow> </m:mrow> </m:math> G=AC , the equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>c</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mi>c</m:mi> <m:mrow> <m:mi mathvariant="normal">Π</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> </m:mrow> </m:math> cx=\varphi(x)c^{\Pi(x)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi>A</m:mi> </m:mrow> </m:math> x\in A , determine a permutation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>φ</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>A</m:mi> </m:mrow> </m:mrow> </m:math> \varphi\colon A\to A and a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Π</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo stretchy="false">→</m:mo> <m:msub> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> \Pi\colon A\to\mathbb{Z}_{n} on 𝐴 characterized by the following properties: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>φ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mn>1</m:mn> <m:mi>A</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow>