A generalized Powers averaging property for commutative crossed products

We prove a generalized version of Powers' averaging property that characterizes simplicity reduced crossed products $C(X) \rtimes_\lambda G$, where $G$ is countable discrete group, and $X$ compact Hausdorff space which acts on minimally by homeomorphisms. As consequence, we generalize results Hartman Kalantar unique stationarity to the state G$ Kawabe's amenable subgroups $\operatorname{Sub}_a(X,G)$. This further lets us result first named author intermediate C*-algebras. if $C(Y) \subseteq C(X)$ an inclusion unital commutative $G$-C*-algebras with minimal simple, then any C*-algebra $A$ satisfying G A C(X) simple.