Finite 2-groups in which the set of self-centralizing abelian normal subgroups with at least three generators is empty ($SCN_3(2)=\varnothing$)

In this paper we study finite 2-groups in which each abelian normal subgroup is metacyclic, i.e. $SCN_3(2)=\varnothing$. The main result: a finite 2-group with $SCN_3(2)=\varnothing$ is an extension of a metacyclic group by a group isomorphic to a subgroup of the dihedral group of order 8.