  Let $G$ be a group of order $n$ and $H$ be a subgroup of order $m$ of $G$.
Denote by $\psi_H(G)$ the sum of element orders relative to $H$ of $G$. It is
known that if $G$ is nilpotent, then $\psi_H(G)\leq\psi_{H_m}(G)$, where $H_m$
is the unique subgroup of order $m$ of $C_n$. In this note, we show that this
inequality does not hold for infinitely many finite solvable groups.
