On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4

In Theorem 2.3 we determine finite 2-groups all of whose minimal nonabelian subgroups are of order 8 (i.e., they are isomorphic to D8 or Q8). In Corollary 2.4 we determine finite 2-groups all of whose minimal nonabelian subgroups are isomorphic and have order 8. In Corollary 2.5 we show that a minimal non-Dedekindian finite 2-group is either minimal nonabelian or is isomorphic to Q16. In further three theorems we classify finite 2-groups all of whose minimal nonabelian subgroups are pairwise isomorphic and have order >8 and exponent 4. This solves some problems stated by Berkovich [Y. Berkovich, Groups of prime power order, Parts I and II (with Z. Janko), in preparation].