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As a consequence, if $G$ is finitely generated, $|G:N|<\\infty$, and all abelian groups $H^{\\mathrm{ab}}$, $N\\subseteq H\\subseteq G$, are torsion free, then $N^{\\mathrm{ab}}$ must be a pseudo permutation module for $G/N$ (cf. Thm. B). From Theorem A one also deduces a non-trivial relation between the order of the transfer kernel and co-kernel which determines the Hilbert-Suzuki multiplier (cf. Thm. C). 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