Classify cyclic-group skew morphisms with automorphic induced quotient

Classify all skew morphisms \(\varphi\) on cyclic groups for which the induced skew morphism \(\overline{\varphi}\) on the quotient by \(\operatorname{Ker}\varphi\) is an automorphism.

Background

The paper uses a covering technique in which a skew morphism is studied through the induced skew morphism on a quotient by its kernel. In the skew-type 4 case, the induced skew morphism on $\mathbb{Z}_n/\operatorname{Ker}\varphi$ is an automorphism of order 2.

The authors suggest that this approach may apply more broadly, but they do not provide a classification for all cyclic-group skew morphisms whose induced quotient skew morphism is an automorphism.