Finite groups in which every proper characteristic subgroup is cyclic

Let $G$ be a finite non-cyclic, non-characteristically simple group with the property that all proper characteristic subgroups of $G$ are cyclic. We call such a group $\mathrm{C}\mathrm{C}\mathrm{S}$ group, short for \emph{Characteristic Cyclic}. In this paper, we provide a complete classification of these groups. As an application of our main result, we also make some progress toward the classification of minimal non-cyclic skew braces.