Let G be a finite group and N be a normal subgroup of G. Denote by \({{\,\textrm{Irr}\,}}(G|N)\) the set of all irreducible complex characters of G whose kernels do not contain N. For \(\chi \in {{\,\textrm{Irr}\,}}(G)\) , the number \({{\,\textrm{cod}\,}}(\chi )=|G:\textrm{ker}(\chi )|/\chi (1)\) is called the co-degree of \(\chi \) . In this paper, we prove that if, for each pair \(\chi , \phi \in {{\,\textrm{Irr}\,}}(G|N)\) with \(\textrm{cod} (\chi )\not = \textrm{cod} (\phi )\) , the greatest common divisor of \(\textrm{cod} (\chi )\) and \(\textrm{cod} (\phi )\) is square-free, then N is solvable.
