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.