When studying character theory (specifically, of normal subgroups), one comes across the concept of the inertia group. If $N \unlhd G$, where $G$ is a finite group, then, $G$ acts on $\operatorname{Irr}(N)$, the group of irreducible characters of $N$, by $\theta^g(x) = \theta(gxg^{-1})$ (which is well-defined by the normality of $N$, and is easily seen to be a character). The inertia group of $\theta$ is then $$I_G(\theta) = \{g \in G \mid \theta^g = \theta\}$$
which is the stabilizer of $\theta$ in this action.
On the seemingly distinct field of algebraic field theory, though, there also exists an inertia group. If $K$ is an algebraic number field, $\mathfrak{p}$ is one of its primes (i.e., prime ideals of its ring of algebraic integers) and $L$ is a Galois extension with group $G = \operatorname{Gal}(L:K)$, then $G$ acts on the primes of $L$ which lie over $\mathfrak{p}$. If $\mathfrak{P}$ is one such prime, we write $G(\mathfrak{P}) = \{\sigma \in G \mid \sigma(\mathfrak{P}) = \mathfrak{P}\}$, which is the decomposition group, and the stabilizer of $\mathfrak{P}$. The inertia group is the subgroup $$I(\mathfrak{P}) = \{\sigma \in G(\mathfrak{P}) \mid \sigma(r) \equiv r \pmod{\mathfrak{P}}, \forall r \in R'\}$$
where $R'$ is the ring of algebraic integers of $L$.
So, both of these groups are point stabilizers in some form, though I can't see any other similarity between the two. I reckon there must be some deep connection, since the terminology is the same, even when it comes to related topics such as ramification indices. In short:
Is there some major connection between the two ideas of inertia groups?
Thanks in advance!
I strongly suspect that the concept of group actions and stabilizers is a bit newer than the first generation of abstract algebra books. My evidence is van der Waerden's Algebra, which does not define the term Stabilizer (though it does e.g. Normalizer).
Number theory's definition of inertia group already occurs in Hilbert's Zahlbericht of 1894.
So it seems natural that in the development of Clifford theory (which is not that much newer than van der Waerden's book) someone noticed the similarity of behavior between characters and their conjugates, and conjugate ideals above a prime, borrowing the term "inertia group" for what we now would call a stabilizer.