In this paper, we classify smooth 5-manifolds with fundamental group
isomorphic to $\z/2$ and universal cover diffeomorphic to $S^2 \times S^3$. As
a consequence, a classification of smooth free involutions on $S^2 \times S^3$
up to conjugation is obtained.
In this paper, a classification of free involutions on 3-dimensional homotopy
complex projective spaces is given. By the $\mathbb Z_2$-equivariant
Montgomery-Yang correspondence, we obtain all smooth involutions on $S^6$ with
fixed-point set an embedded $S^3$.