A remark on the freeness condition of Suzukis correspondence theorem for intermediate ${rm C}^ast$-algebras

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 Suzukis theorem that proves the statement for free actions.