Let $ G $ be a finite group and $ N $ be a normal subgroup of $ G $. Let $ G/N $ have two irreducible characters of degrees $ p\_1$ and $ p\_2$, where $ p\_1$ and $ p\_2$ are different primes. Let $ G/N $ have no irreducible character such that $ p\_1p\_2\mid \chi (1) $. If $ (p\_1p\_2, |N|)=1$, can we say that $G $ has no irreducible character $\chi $ such that $ p\_1p\_2\mid \chi (1) $?
I guess that this is impossible but I could not find any counterexample for it.



A question about relation of the character degrees of $ G/N $ and $ G $
It is possible for $G$ to have such a character. The smallest example has order $120$. Let $N = \langle k \rangle \cong C\_5$. Let $G = N \rtimes S\_4$ where the action of the symmetric group $S\_4$ on $N$ is non-trivial but factors through the sign representation. Thus $k^{(12)} = k^{-1}$, $k^{(234)} = k$ and $\langle k, (12)\rangle$ is dihedral of order $10$. The character degrees of $G$ are $\{1,2,3,6\}$ and the character degrees of $G/N \cong S\_4$ are $\{1,2,3\}$. 


