Finite groups with a large normalized sum of element orders

For a finite group $G$, let $\psi (G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as ${\psi }^{\mathrm{\prime }}(G):=\psi (G)\mathrm{/}\psi ({\mathcal{C}}_{\mathrm{\mid }G\mathrm{\mid }})$, where ${\mathcal{C}}_{\mathrm{\mid }G\mathrm{\mid }}$ is the cyclic group of the same order as $G$. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if ${\psi }^{\mathrm{\prime }}(G)>{\psi }^{\mathrm{\prime }}(D_8)=\frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying ${\psi }^{\mathrm{\prime }}(G)>{\psi }^{\mathrm{\prime }}(A_4)=\frac{31}{77}$, thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi [Canad. Math. Bull. 65 (2022), 30–38], and thus providing a more complete answer to a corresponding conjecture of Tǎrnǎuceanu.