Writing this matrix expression in terms of vec operator

Question

Let $X$ be a $p \times k $ matrix with $p > k.$ Is there a natural way to write either $$\operatorname{Tr}\left\{C^T X(X^T X)^{-1/2}\right\}$$ or $$\operatorname{Tr}\left\{B\left[X(X^TX)^{-1/2}\right]^T A \left[X(X^TX)^{-1/2}\right] \right\}$$ in terms of $\operatorname{vec} X,$ where vec is the operator described here? $C$ has the same dimensions as $X$ while $A$ and $B$ are symmetric $k \times k$ matrices. If it's helpful, one may also assume $B$ is diagonal.