Finite $2$-groups with a non-Dedekind non-metacyclic norm of Abelian non-cyclic subgroups

The authors study finite $2$-groups with non-Dedekind non-metacyclic norm $N_{G}^{A}$ of Abelian non-cyclic subgroups depending on the cyclicness or the non-cyclicness of the center of a group $G$. The norm $N_{G}^{A}$ is defined as the intersection of the normalizers of Abelian non-cyclic subgroups of $G$. It is found out that such $2$-groups are cyclic extensions of their norms of Abelian non-cyclic subgroups. Their structure is described.