Can't we define them as the subgroups of the automorphism group of an arbitrary extension $L/K$ that fixes a prime $Q$ in $\mathcal O_L$ over a prime $P$ in $\mathcal O_K$?
An ideal answer would probably point out some irredeemable flaw that renders the concept trivial/useless. One problem I can see is that $D/I$ might not be surjective onto the galois group of $\mathcal O_L/Q$ but I am not sure if this is fixable or not...