We prove a generalized version of Powers’ averaging property that characterizes simplicity of reduced crossed products
C
(
X
)
⋊
λ
G
C(X) \rtimes _\lambda G
, where
G
G
is a countable discrete group, and
X
X
is a compact Hausdorff space which
G
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
)
⋊
λ
G
C(X) \rtimes _\lambda G
and to Kawabe’s generalized space of amenable subgroups
Sub
a
(
X
,
G
)
\operatorname {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
)
⊆
C
(
X
)
C(Y) \subseteq C(X)
is an inclusion of unital commutative
G
G
-C*-algebras with
X
X
minimal and
C
(
Y
)
⋊
λ
G
C(Y) \rtimes _\lambda G
simple, then any intermediate C*-algebra
A
A
satisfying
C
(
Y
)
⋊
λ
G
⊆
A
⊆
C
(
X
)
⋊
λ
G
C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G
is simple.