  Let $X$ be a finitistic space with the mod 2 cohomology of the product space
of a projective space and a 4-sphere. Assume that $X$ admits a free involution.
In this paper we study the mod 2 cohomology algebra of the quotient of $X$ by
the action of the free involution and derive some consequences regarding the
existence of $\mathbb{Z}_2$-equivariant maps between such $X$ and an
$n$-sphere.
