In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere bundles over $S^4$. Also, Kervaire and Milnor proved that there are exactly 28 h-cobordism (therefore diffeomorphism) classes of homotopy spheres in dimension 7.

Does every class contain an exotic sphere arising as the total space of an $S^3$-bundle over $S^4$?, if not, how can one determine the number of classes with such representatives?, how do they look like?