Finite groups with large sum of element orders compared to cyclic groups

For a finite group $G$ let $\psi (G)$ denote the sum of the orders of elements in $G$. If $G$ has order $n$ then, by a result of Amiri, Jafarian Amiri and Isaacs, $\psi (G)\le \psi (C_n)$, where $C_n$ is the cyclic group of order $n$. We determine all finite groups such that $\psi (G)\ge \frac{1}{3}\psi (C_n)$.