The sum of orders of elements in nonabelian groups of odd order

Denote by $\psi(G)$ the sum of the orders of the elements of a finite group $G$. We obtain an exact upper bound for $\psi(G)$ on the set of nonabelian groups of given odd order $n$ in terms of the minimal prime divisor of $n$. We also describe the finite groups on which this bound is achieved.