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This is a continuation of the paper “Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen”, [6]. In [6], the classes $$\mathfrak{A}$$ and $$\mathfrak{D}$$ were introduced. We give sufficient conditions to conclude thatG is in $$\mathfrak{D}(\mathfrak{A} \cap \mathfrak{D})$$ if one knows thatG/G 0 is in $$\mathfrak{D}(\mathfrak{A} \cap \mathfrak{D})$$ . If a groupG is in $$\mathfrak{A} \cap \mathfrak{D}$$ and ifG satisfies the Chu-duality then all closed subgroups ofG satisfy the Chu-duality. The Chu-quasi-dual of the Heisenberg groupH with integral coefficients is computed. It is shown thatH does not satisfy the Chu-duality, thatH is in $$\mathfrak{A}$$ , and thatH is not in $$\mathfrak{D}$$ .