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 $\psi'(G) := \psi(G)/\psi(\mathcal{C}_{|G|})$, where $\mathcal{C}_{|G|}$ 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 $\psi'(G)>\psi'(D_8) = \frac{19}{43}$, 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 $\psi'(G)> \psi'(A_4) = \frac{31}{77}$, 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 T\v{a}rn\v{a}uceanu.