  Let G be a finite group and let k be a field. We say that G is a projective
basis of a k-algebra A if it is isomorphic to a twisted group algebra k^\alpha
G for some class \alpha in H^2(G,k^\times), where the action of G on k^\times
is trivial. In a preceding paper by Aljadeff, Haile and the author (Projective
bases of division algebras and groups of central type, Israel J. Math. 146
(2005) 317-335) it was shown that if a group G is a projective basis in a
k-central division algebra then G is nilpotent and every Sylow-p subgroup of G
is on the short list of families of p-groups, denoted by \Lambda. In this paper
we complete the classification of projective bases of division algebras by
showing that every group on that list is a projective basis for a suitable
division algebra. We also consider the question of uniqueness of a projective
basis of a k-central division algebra. We show that basically all groups on the
list \Lambda but one satisfy certain rigidity property.
