It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by applying the local norm map at each place, with image $N_{L/K}\in C_K$, which is a closed subgroup and corresponds to an abelian extension etc. etc.
However, there is also a closed embedding $C_K\to C_L$ since $\mathbb{A}_L\cong L\otimes \mathbb{A}_K$ (plus a little more work with general topology/global fields). Hence actually $N_{L/K}\le C_K \le C_L$, so $N_{L/K}$ is a closed subgroup of $C_L$ which is also its image under some quotient. Hence $0\to N_{L,K} \to C_L \to C_L/N_{L,K} \to 0$ actually splits, i.e. $C_L \cong N_{L,K} \oplus C_L/N_{L,K}$. I think everything still applies if we pass to some modulus too.
This seemed kind of weird, but the prospect of working out the consequences of an actual idelic computation were daunting, so I thought I'd ask here. The third terms resembles one side of the Artin reciprocity. Is there a story here? What significance does this have?