Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. We consider the skew products on $\mathbb {T} \times {\Gamma \backslash G}$ and prove that the Möbius function is linearly disjoint from these skew products which improves the recent result of Huang, Liu and Wang [‘Möbius disjointness for skew products on a circle and a nilmanifold’, Discrete Contin. Dyn. Syst. 41(8) (2021), 3531–3553].

In this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.