  Let $G$ be a finite group and $\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$,
where $o(a)$ denotes the order of $a$ in $G$ and $\exp(G)$ denotes the exponent
of $G$. Under a natural hypothesis, in this note we determine the groups $G$
such that $\forall\, H,K\leq G$, $H\subseteq K$ implies
$\varphi(H)\mid\varphi(K)$. This partially answers Problem 5.4 in \cite{6}.
