$\DeclareMathOperator\GL{GL}$Can you describe the maps from $\GL(n, \mathbb{R})$ to $\GL(n, \mathbb{R})$ that are equivariant w.r.t. right multiplication by $\GL(n, \mathbb{Z})$? I'm interested even in classes of examples, not necessarily a full description.

Of course, there are maps that are equivariant to the whole $\GL(n, \mathbb{R})$. What else is there?

Thank you for your help.