Here is an interesting problem which arose out of a discussion:
Let $a, m$ be two coprime natural numbers and let $P$ denote the set of prime integers satisfying $p \equiv a \pmod{m}$. Does there exist a (finite) Galois extension $K / \mathbb{Q}$ and a normal subgroup $H$ of the Galois group $G:=Gal(K / \mathbb{Q})$ such that $|H|/|G| = 1/\phi(m)$ (where $\phi$ denotes the Euler Totient function) and for all sufficiently large integer primes $p$, we have $p \in P \iff \sigma_p \in H$?
Here $\sigma_p$ denotes the conjugacy class of the Frobenius substitution or Frobenius element (also known as the Frobenius Conjugacy Class), which is defined for all integer primes $p$ (not ramifying in $K$) and for all primes $\mathfrak{p}$ of $K$ lying over $p$, as the inverse image of the Frobenius automorphism under the canonical isomorphism $\pi_{\mathfrak{p}}: D_{\mathfrak{p}} \rightarrow Gal(k(\mathfrak{p})/\mathbb{F}_p)$ to the Galois Group of residue field extensions).
If we relax the condition of $H$ being a normal subgroup to that of $H$ simply being a subset that is closed under conjugation by elements of $G$, then considering the cyclotomic extension $K:=\mathbb{Q}_m = \mathbb{Q}(\zeta)$ (where $\zeta$ is of course a primitive $m$-th root of unity) does our job: it is an Abelian extension so that every subset of $G = Gal(K/\mathbb{Q})$ is closed under conjugation and it turns out that for every integer prime $p$, we have the equivalence $$p \in P \iff \sigma_p(\zeta) = \zeta^p = \zeta^a$$ (since the Frobenius element in a cyclotomic extension is simply the $p$-th power map on a generator). Therefore, letting $H$ be the singleton consisting of the $\mathbb{Q}$-automorphism of $K$ determined by $\sigma(\zeta) := \zeta^a$ does our job (since $|G|=\phi(m)$).
Of course $H$ (as defined above) despite being closed under conjugation is not a subgroup of $G$, and I can't seem to obtain any $G$ and $H$ which do the job either, so I am starting to suspect that the statement might be false. Any help would be appreciated, thanks.
Addendum: The conditions imposed on $P$ might be strongly reminiscent of the Chebotarev Density Theorem, and in fact, if we relax the condition of $H$ being a subgroup of $G$ to simply that of being a subset closed under conjugation then we do get the conditions of Chebotarev. I would really appreciate some references or resources where such subsets $S$ of the primes (namely, those satisfying the condition that there exists $K, H$ as above such that $H$ is normal in $G$ and for every large enough prime $p$, $\sigma_p \in H \iff p \in S$) are studied? Perhaps they have some additional interesting properties (besides those arising out of Chebotarev)? Do there exist necessary and sufficient conditions characterizing them?
Edit: Ok, so just a clarification based on a comment. I am looking for a finite Galois extension $K/\mathbb{Q}$ such that there is some normal subgroup $H$ of $G$ which satisfies, for all sufficiently large integer primes $p$, the equivalence $p \in P \iff \sigma_p \in H$ (In other words, is there some finite Galois extension $K/\mathbb{Q}$ and some normal extension $E$ of $\mathbb{Q}$ contained in $K$ such that for all sufficiently large primes $p$, the Frobenius conjugacy class $\sigma_p$ fixes $E$ pointwise exactly for those $p \in P$?)