Connection between ramification in number fields and Clifford theory

It seems not too mysterious, since the statements about prime ideals can be recast into the language of representation theory. Given a commutative ring $R$, the spectrum of prime ideals corresponds to the irreducible $R$-representations. That is to say, the simple $R$-modules are precisely the quotients $R/P$ where $P$ is a prime ideal.

So it seems likely that a general statement of Clifford's theorem would capture the number theory situation.

I can take you part of the way there: In Clifford theory, I'm pretty sure that you can replace the field $K$ with a commutative ring $R$, and talk about $R[G]$- and $R[N]$-modules, and basically the same Clifford theorem holds (see Clifford theory - Encyclopedia of Mathematics).

In particular, if $R$ is $\mathcal{O}_K$, $G$ is the Galois group of $L/K$ and $N$ is the trivial subgroup, we get a Clifford theory statement about $\mathcal{O}_K[G]$-modules and simple $\mathcal{O}_K$-modules (which correspond to the prime ideals of $\mathcal{O}_K$).

Then when $\mathcal{O}_L$ has an integral normal basis, it is a rank 1 free $\mathcal{O}_K[G]$-module, so the Clifford theory statement would seem to apply here.

I suppose then some modest generalization to Clifford's theorem can be made to accommodate the cases when $\mathcal{O}_L$ does not have an integral normal basis.