Operators preserving the representation of a semidirect product group

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