Finite groups with a small proportion of vanishing elements

The function ${\mathrm{P}}_{\mathbf{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes (as proved in [12]) only values $\frac{m-1}{m}$, for $1\le m\le 6$, in the interval $[0,{\mathrm{P}}_{\mathbf{v}}(A_7))$.In this paper, we give a complete classification of the finite groups $G$ such that ${\mathrm{P}}_{\mathbf{v}}(G)=\frac{m-1}{m}$ for $m=1,2,\cdots ,6$.