Cohomology algebra of orbit spaces of free involutions on the product of projective space and 4-sphere

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.