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?