Footnote to a theorem of Phagan on actions of a cyclic group

We provide a simplified proof of a recent theorem of Phagan (arXiv:2509.14083) relating the number of orbits of a given length of an action $G\curvearrowright S$ of a cyclic group with the number of orbits of the induced action $H\curvearrowright S$ of a subgroup $H\subseteq G$. This combinatorial fact has applications to notions of arithmetic similarity of number fields, as investigated by Phagan (op. cit.).