= Solution
The matrix-element orthogonality theorem for irreducible unitary representations states
$$
\sum_{g\in G}
\overline{M_{\alpha,ij}(g)}M_{\beta,kl}(g)
=\frac{|G|}{d_\alpha}
\delta_{\alpha\beta}\delta_{ik}\delta_{jl},
$$
and $\sum_\alpha d_\alpha^2=|G|$. Consequently the normalized vectors
$$
|\alpha ij\rangle
=\sqrt{\frac{d_\alpha}{|G|}}
\sum_{g\in G}\overline{M_{\alpha,ij}(g)}|g\rangle
$$
form an orthonormal basis. The non-abelian <quantum Fourier transform> is the unitary basis change
$$
\boxed{|\alpha ij\rangle\longmapsto|\alpha\rangle|i\rangle|j\rangle},
$$
up to the harmless choice of transform direction and complex-conjugation convention.
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