  A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any
subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element
orders of $H$ and $G$, respectively. In this paper, we extend a result provided
in [10], by classifying the finite groups whose all subgroups are
$\psi$-divisible. Since the existence of $\psi$-divisible groups is related to
the class of square-free order groups, we also study the sum of element orders
and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the
concept of $\psi$-normal divisible group, i.e. a group for which the
$\psi$-divisibility property is satisfied by all its normal subgroups. Using
simple and quasisimple groups, we are able to construct infinitely many
$\psi$-normal divisible groups which are neither simple nor nilpotent.
