On a group-theoretical generalization of the Euler’s totient function

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