On some classes of smooth transformations in the space of symmetric matrices

Let $\mathrm{Sym}(n)$ be the space of $n$-dimensional real symmetric matrices. Two families of infinitely smooth transformations in $\mathrm{Sym}(n)$ are considered. First, the family of transformations
$$ {\mathcal F}\colon\operatorname{Sym}(n)\to\operatorname{Sym}(n), $$
having the following property: for any matrix $X\in\mathrm{Sym}(n)$ and an orthogonal matrix $C$ such that $C^{-1}XC$ is a diagonal matrix, $C^{-1}\mathcal{F}(X)C$ is also a diagonal matrix. Second, the family of transformations
$$ {\mathcal G}\colon\operatorname{Sym}(n)\to\operatorname{Sym}(n), $$
such that the diagonal entries of the matrix $C^{-1}\mathcal{G}(X)C$ are zero whenever the matrix $C^{-1}XC$ is diagonal.