  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
$M$.
