Finite groups with some bounded codegrees

For a character $χ$ of a finite group $G$, the number cod$(χ):=|G:\mathrm{ker}(χ)|/χ(1)$ is called the codegree of $χ$.In this paper, we give a solvability criterion for a finite group $G$ depending on the minimum of the ratio $χ(1)^2 /\mathrm{cod}(χ)$, when $χ$ varies among the irreducible characters of $G$.