Inverse of the product of non-square, full-rank matrices

Question

I have a matrix $M \in \mathbb{R}^{m \times m}$ defined as: $$ M^{-1} = P^T A P $$ Where $A \in \mathbb{R}^{n \times n}, P \in \mathbb{R}^{n \times m}, n > m$ and both $A$ and $P$ are full rank. Also, $P^T P = I_m$ where $I_m$ is the identity matrix of size $m \times m$. Is there any way to express $M$ in terms of $A^{-1}$? Maybe using the psudoinverse of $P$ or something?