Some density results involving the average order of a finite group

Let $o(G)$ be the average of the element orders of a finite group $G$. A research topic concerning this quantity is understanding the relation between $o(G)$ and $o(H)$, where $H$ is a subgroup of $G$. Let $\mathcal{N}$ be the class of finite nilpotent groups and let $L(G)$ be the subgroup lattice of $G$. In this paper, we show that the set $\{\frac{o(G)}{o(H)}\mathrm{\mid }G\in \mathcal{N},H\in L(G)\}$ is dense in $[0,\mathrm{\infty })$. Other density results are outlined throughout the paper.