Maximization of $tr(Q^T S^2 Q (Q^T S Q)^{-1})$ with respect to $Q$ for a given $S$

Question

I have a $n \times n$ positive definite matrix $S$, suppose $Q$ is a $n \times k$ unknown matrix with $k <n$. I have an objective function of the form $J=tr(Q^T S^2 Q (Q^T S Q)^{-1} )$. I know that the solution of the maximization of $tr(Q^T S Q (Q^T Q)^{-1} )$ with respect to $Q$ is given by $Q$ consisting of the eigenvectors corresponding to the largest $k$ eigenvalues of $S$. Now, the maximization of $J=tr(Q^T S^2 Q (Q^T S Q)^{-1} )$, what's the solution of $Q$ here? Do I have the same matrix of eigenvectors? Thanks a lot!