Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field uniquely identifies the Galois group of (the Galois closure over $\mathbb{Q}$ of) that field. Of course not always: there are $S_n$-extensions whose discriminant is the same as that of some quadratic field; on the other hand, certain small discriminants (such as -3, which must belong to the group $C_2$) can only occur once by standard discriminant bounds. Intuitively, I would think that such unique discriminants are pretty common, so:
Q1: Is there a reasonable heuristic for the asymptotic of, e.g., number of degree-$n$ number fields whose discriminant suffices to identify the Galois group? Is it even clear that such fields exist for all $n$?
I was also surprised by the following easy example: If $F/\mathbb{Q}$ is a tamely ramified quartic $A_4$-extension, then its discriminant can NEVER identify the Galois group! Namely, this discriminant would have to be a square; if it involves more than one prime factor, then some $C_2\times C_2$-extension has the same discriminant; but if it has only one prime factor, then this prime must be $\equiv 1$ mod $3$, and so there is a $C_3$-extension with the same discriminant!
Q2: Are there other examples as this, maybe infinite families, of permutation groups $G$ that can never be recognized from the discriminant of a (possibly, tame) $G$-extension?
