How large a union of proper subgroups of a finite group can be?

Let $k$ be a positive integer and $G$ be a finite group that cannot be written as the union of $k$ proper subgroups. In this short note, we study the existence of a constant $c_k\in (0,1)$ such that $|\cup_{i=1}^k H_i| \leq c_k|G|$,forallpropersubgroups$H_1$,\mathrm{.}\mathrm{.}\mathrm{.},$H_k$of$G$.