  Given a number field $k$ and a finitely generated subgroup $\mathcal{A}
\subseteq k^*$, we study the distribution of $S_4$-quartic extensions of $k$
such that the elements of $\mathcal{A}$ are norms. We show that the density of
such extensions is the product of so-called "local masses" at every place of
$k$. We give these local masses explicitly in almost all cases and give an
algorithm for computing the remaining cases.
