Finite 2-groups all of whose maximal cyclic subgroups of composite order are self-centralizing

We determine here the structure of the groups of the title which are of exponent at least 4. It turns out that such a group is either cyclic or of maximal class (i.e., dihedral, semidihedral or generalized quaternion) or a uniquely determined group of order 2 5 (Theorem 1.1). This solves Problem no. 523 in Berkovich [Y. Berkovich. Groups of prime power order . In preparation.] for p = 2. The corresponding problem for p > 2 is open but very difficult.