\(CZ\)-groups with nonabelian normal subgroup of order \(p^4\)

A \(p\)-group \(G\) with the property that its every nonabelian subgroup has a trivial centralizer (namely only its center) is called a \(CZ\)-group. In Berkovich's monograph (see [1]) the description of the structure of a \(CZ\)-group was posted as a research problem. Here we provide further progress on this topic based on results proved in [5]. In this paper we have described the structure of \(CZ\)-groups \(G\) that possess a nonabelian normal subgroup of order \(p^4\) which is contained in the Frattini subgroup \(\Phi(G).\) We manage to prove that such a group of order \(p^4\) is unique and that the order of the entire group \(G\) is less than or equal to \(p^7\), \(p\) being a prime. Additionally, all such groups \(G\) are shown to be of a class less than maximal.