This paper gives the cohomology classification of finitistic spaces X equipped with free actions of the group G = S 3 and the cohomology ring of the orbit space X/G is isomorphic to the integral cohomology quaternion projective space HP n. We have proved that the integral cohomology ring of X is isomorphic either to S 4n+3 or S 3 × HP n. Similar results with other coefficient groups and for G = S 1 actions are also discussed. As an application, we determine a bound of the index and co-index of cohomology sphere S 2n+1 (resp. S 4n+3) with respect to S 1-actions (resp. S 3-actions).