Inverse of quadratic form with rectangular matrix

Question

Suppose that $A \in \mathbb{R}^{n \times r},$ where $r<n$, where the columns of $A$ are orthonormal. Now suppose that $X \in \mathbb{R}^{n \times n}$ is an invertible matrix. Is there an explicit formula for $(A^T X A)^{-1}$, perhaps in terms of the pseudo-inverse of $A$? Perhaps more importantly, I care about the quantity $A (A^T X A)^{-1} A^T$, where the columns of $A$ is part of an orthonormal basis of $\mathbb{R}^n.$