  Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$.
In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is
supersolvable. In particular, some new characterizations of the well-known
groups $\mathbb{Z}_2\times\mathbb{Z}_4$ and $A_4$ are obtained. We also show
that $\sigma_1(G)<c$ does not imply the supersolvability of $G$ for no constant
$c\in(2,\infty)$.
