Properties of a matrix $G$ where $GA=BG$ if $A$ and $B$ are orthogonal matrices

Question

What are the properties of an $n\times n$ matrix $G$ if:

$GA=BG \space\space\space \space\space\space \space\space\space A,B\in O(n) $

I suspect that $G$'s columns must be mutually orthogonal, but cannot proved so.

Edit: For particular matrices $A,B$, which are fixed with respect to $G$.

Answer 1

Taking $A=B=I_n$, where $I_n$ is the $n\times n$-identity matrix, we see that we need to impose some condition on $(A,B)$ to have interesting results.

Please let me know if you meant: "For all orthogonal matrices $A$, $B$." Edit: In that case take $A=I_n$, $B=-I_n$ and infer $G=0$ (real entries).