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?
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 grouptheoretic 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 GruenbergWeiss, which gives an impressively vast generalization of the grouptheoretic (and hence the idealtheoretic) principal ideal theorem entirely in terms of kernels of related transfer maps. 

