is $L^2(M\ltimes G)\cong \oplus_G L^2(M)$ as Hilbert spaces

Question

Let $M$ be a $II_1$ factor then I want to show that if $G$ is a finite group acting on $M$ then considering the GNS of the crossed product with respect to the trace one has the unitary isomorphism $L^2(M\ltimes G)\cong \oplus_G L^2(M)$. It seems plausible as each $mg$ is orthogonal to $nh$ for $h$ not equal to $g$. My problem is that the inner product on the right hand side is given by the trace but the left had side if we take $\langle mg,ng\rangle=\tau(m\alpha_g(n))$ which I don't see how it should be equal to $\tau(mn)$ am I missing something? This is an attempt to show that the index for $M$ in $M\ltimes G$ is $|G|$.