Injectivity of Transfer (Verlagerung) map  Let $ K $ be a finite extension of a $p$-adic field or a number field, L a finite extension of $K$. The following fact holds: $
 \text{Gal}(K^{\text{ab}} / K) \rightarrow \text{Gal}(L^{\text{ab}} / L) 
$, where the arrow is the Transfer (Verlagerung) map, is  injective.
I wonder whether this is an arithmetic fact or a fact about absolute Galois group of fields in general?
 A: I'm not sure how to answer the more philosophical question (it's likely you could encode enough of the axioms to force the purely group-theoretic version of the question to be true, but to ask whether that's what's "really" going on....), but it's certainly not true for all pairs of groups that fit in a similar commutative diagram -- in fact, it's not even true for all such pairs of groups coming from similar questions in algebraic number theory.  For example, instead of taking the maximal abelian extension of $K$, take the maximal abelian extension of $K$ which is unramified outside of a set of primes, or split completely at a set of primes, or both -- and you'll pick up a kernel to you transfer map (see Gras, Class Field Theory for some specific calculations of kernels like this).  A very relevant related topic worth bringing up is the theorem of Gruenberg-Weiss, which gives an impressively vast generalization of the group-theoretic (and hence the ideal-theoretic) principal ideal theorem entirely in terms of kernels of related transfer maps.
