A density result on the sum of element orders of a finite group

Let $\mathcal{G}$ be the class of all finite groups and consider the function ${\psi }^{\mathrm{\prime }\mathrm{\prime }}:\mathcal{G}⟶(0,1]$, given by ${\psi }^{\mathrm{\prime }\mathrm{\prime }}(G)=\frac{\psi (G)}{\mathrm{\mid }G{\mathrm{\mid }}^2}$, where $\psi (G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of ${\psi }^{\mathrm{\prime }\mathrm{\prime }}$ is a dense set in $[0,1]$. Also, we study the injectivity and the surjectivity of ${\psi }^{\mathrm{\prime }\mathrm{\prime }}$.