Given symmetric $A,B\in\Bbb R^{n\times n}$ what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$AX=XB$$ holds?

**(1)** What if $A,B,X\in\Bbb F_{p^r}^{n\times n}$ is considered?

Note: $X'$ means the transpose of $X$.

**(2)** Is there a test to see if there is no such $X$?

**(3)** Suppose we know one such $X$ we can find all such $X$ from finding $U$ such that $UU'=I$ and $UA=AU$ or $BU=UB$. Do all such $X$ arise this way?