Comparing the density of $D_4$ and $S_4$ quartic extensions of number fields

When ordered by discriminant, it is known that about 83% of quartic fields over $\mathbb {Q}$ have associated Galois group $S_4$, while the remaining 17% have Galois group $D_4$. We study these proportions over a general number field $F$. We find that asymptotically 100% of quadratic number fields have more $D_4$ extensions than $S_4$ and that the ratio between the number of $D_4$ and $S_4$ quartic extensions is biased arbitrarily in favor of $D_4$ extensions. Under Generalized Riemann Hypothesis, we give a lower bound that holds for general number fields.