Finite nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4

We classify here the title groups and note that such groups must be of exponent >4 if both D8 and H2=〈a,b|a4=b4=1,ab=a−1〉 appear as subgroups (Theorem 1.1). This solves a problem stated by Y. Berkovich in [Y. Berkovich, Groups of Prime Power Order, I and II (with Z. Janko), Walter de Gruyter, Berlin, 2008]. On the other hand, if G is a nonabelian finite 2-group all of whose minimal nonabelian subgroups are non-metacyclic and have exponent 4, then G must be of exponent 4 (Theorem 1.5). We also solve a more general problem Nr. 1475 of Berkovich [Y. Berkovich, Groups of Prime Power Order, I and II (with Z. Janko), Walter de Gruyter, Berlin, 2008] by classifying nonabelian finite 2-groups of exponent e2 (e⩾3) which do not have any minimal nonabelian subgroup of exponent e2 (Theorem 1.6). Finally, we prove Lemma 1.7 which might be useful for future investigations.