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

For a finite group $G$ let $ψ(G)$ 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, $ψ(G)\leψ(C_n)$, where $C_n$ is the cyclic group of order $n$. We determine all finite groups such that $ψ(G)\ge\frac{1}{3}ψ(C_n)$.