Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know that $N$ is a normal subgroup of $G$ such that $(p_1p_2,|N|)=1$. Can we say that always there exists an irreducible character $\eta$ in $Irr(G/N)$ such that $p_1p_2\mid \eta(1)$? Thanks for your helps.
The answer is 'no'. A counterexample is given by $G=C_7\rtimes C_6$ with faithful action. If you induce any non-trivial character from $C_7$, the induction will be 6-dimensional irreducible. But of course there is no irreducible 6-dimensional character of $G/C_7\cong C_6$.