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