We extend the theory of quasi-invariant states for compact group actions on \(C^{*}\) -algebras to the setting of semidirect product groups. Given a compact semidirect product \(K=G\rtimes _{\phi }H\) , where \(\phi \) is a continuous homomorphism from H into \(\operatorname {Aut}(G)\) , we characterize actions of K on \(C^{*}\) -algebras in terms of compatible actions of the component groups G and H. We establish the fundamental properties of K-quasi-invariant states, including cocycle identities, lifting to von Neumann algebras, averaging properties, and prove the main result that under appropriate modular commutation conditions, the GNS representation of a quasi-invariant state is unitarily equivalent to that of its averaged state. This generalizes the framework established by Griseta [3] for single compact groups.
