  It is a long-standing open problem raised by Starostin to describe all finite
groups with soluble centralizers of involutions. One can observe that if the
centralizer fusion system of an involution is nilpotent, then the centralizer
of that involution is soluble. In this paper, we classify the cases when the
centralizer fusion system of a central involution in a finite group whose all
involutions have soluble centralizers is a nilpotent fusion system. Indeed, we
analyse the case when the solvable radical has odd order and the corresponding
factor group is simple.
