  Let $\Gamma$ be a discrete group satisfying the approximation property (AP).
Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map
which is injective on the non-free part. We prove the one-to-one correspondence
between intermediate ${\rm C}^\ast$-algebras of $C_0(X) \rtimes_r \Gamma
\subset C_0(Y) \rtimes \Gamma$ and intermediate $\Gamma$-${\rm
C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of
Suzuki's theorem that proves the statement for free actions.
