Relate the inverse of $U^\top A^{-1} U$ to $A$

Question

Let $A$ be $N\times N$ symmetric positive definite, and let $U$ be $N\times (N-1)$ with full column rank, i.e. $U^\top U$ invertible. $U^\top A^{-1} U$ is invertible according to this answer, but is there any way to relate $(U^\top A^{-1} U)^{-1}$ to $A$ ? I tried the Sherman-Morrison formula, but I did not manage there.