The present authors constructed in [J. Reine Angew. Math. 455, 183-220 (1994)] inhomogeneous Einstein metrics of positive scalar curvature on compact simply connected 3-Sasakian manifolds \((S(p),g(p))\) in dimension \(4n-5\) for all \(n\geq 3\). The main result here states that \(S(p)\) is either strongly inhomogeneous or \(S(p)\) is homotopy equivalent to the quotient of the unitary group \(U(n)\) by \(U(n-2) \times S^1\), where the action is given by a certain formula. As a corollary, it follows that for \(n>2\), there are countable subfamilies of pairwise homotopy distinct \(S(p)\) manifolds, each of which is a \((4n-5)\)-dimensional compact simply connected, strongly inhomogeneous Einstein manifold of positive scalar curvature.