Operators preserving the representation of a semidirect product group

For a locally compact abelian group $\textbf{R}$ with a uniform lattice $\Lambda$ and a group $G$ that acts on $\textbf{R}$ by continuous automorphisms, we study operators commuting with the representation of $G \rtimes \Lambda$ on $L^2(\textbf{R})$. As a consequence, we give a characterization of shift-dilation preserving operators.