For a Galois extension $L/K$ of algebraic number fields with rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$ respectively, a prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ unramified over $\mathcal{O}_L$, some choice of prime ideal $\mathfrak{P} \subset \mathcal{O}_L$ above $\mathfrak{p}$, and the decomposition subgroup $D_{\mathfrak{P}}<\textrm{Gal}(L/K)$, one can use the following isomorphism: $$ D_{\mathfrak{P}} \cong \textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p})) \tag{$*$} $$ to show that there exists a unique automorphism (the Frobenius automorphism) $\varphi_{\mathfrak{P}} \in \textrm{Gal}(L/K)$ satisfying: $$ \varphi_{\mathfrak{P}}(x) \equiv x^q\ (\textrm{mod}\ \mathfrak{P}) \quad \textrm{for all} \quad x \in \mathcal{O}_L, \tag{$\dagger$} $$
where $q:= \lvert \mathcal{O}_K/\mathfrak{p} \rvert$ is the order of the base residue field.
This is true for all primes $\mathfrak{p} \subset \mathcal{O}_K$ unramified over $\mathcal{O}_L$.
Now I have read somewhere (E. Kowalski, An Introduction to the Langlands Program, p. 9) that when we have a prime $\mathfrak{p} \subset \mathcal{O}_K$ that is ramified over $\mathcal{O}_L$, we have not a single Frobenius element $\varphi_{\mathfrak{P}} \in \textrm{Gal}(L/K)$ satisfying $(\dagger)$, but rather a conjugacy class in $\textrm{Gal}(L/K)$ of such elements.
I have been trying to find resources that treat the existence of the Frobenius conjugacy class for the ramified primes, but have been unable to find any.
The problem I am having is that for ramified primes, we have not the isomorphism $(*)$, but rather: $$ D_{\mathfrak{P}}/I_{\mathfrak{P}} \cong \textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p})), $$
where $I_{\mathfrak{P}} \vartriangleleft D_{\mathfrak{P}}$ is the (non-trivial) inertia subgroup. Hence we cannot view $\textrm{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p}))$ as a subgroup of $\textrm{Gal}(L/K)$ as we can with the unramified primes.
I would much appreciate it if anyone could provide a reference to a resource that treats the Frobenius automorphism for ramified primes, or provide some guidance in resolving my difficulties.
Many thanks.