Smooth structures on a fake real projective space

We show that the group of smooth homotopy 7-spheres acts freely on the set of smooth manifold structures on a topological manifold M which is homotopy equivalent to the real projective 7-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective 7-space. We also show that M has, up to diffeomorphism, exactly 28 distinct differentiable structures with the same underlying PL structure of M and 56 distinct differentiable structures with the same underlying topological structure of M.