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I am no expert, but let me share an idea. For a number field $M$ let us denote by $C_M$ the idele class group of $M$. By class field theory, $K' \subset L'$ is the same as $N_{L'/K}(C_{L'})\subset N_{K'/K}(C_{K'})$, where $N$ stands for the norm map. Let us denote by $U$ the open subgroup $N_{L'/L}(C_{L'})$ of $C_L$, and by $j$ the natural injection of $C_K$ into $C_L$. Then, by the transfer theorem, the previous relation can be rewritten as $N_{L/K}(U)\subset j^{-1}(U)$, that is, $j(N_{L/K}(U))\subset U$. For certain open subgroups $U$ of $C_L$ this relation follows from ramification theory, and further local analysis might extend it to more general open subgroups.

EDIT: I think now that the conclusion $K' \subset L'$ fails when $L$ has two places $w$ and $w'$ above the same place $v$ of $K$ such that $L'/L$ is much more ramified at $w$ than at $w'$. Indeed, viewing $U$ as a subgroup of the ideles $J_L$, this assumption implies that $U_w:=U\cap L_w^\times$ is much smaller than $U_{w'}:=U\cap L_{w'}^\times$. However, $j(N_{L/K}(U))\subset U$ would imply that $N_{L_{w'}/K_v}(U_{w'})\subset U_w$ which is false when $U_w$ is sufficiently small in terms of $U_{w'}$. Perhaps this argument can be rewritten in terms of the transfer map and ramification theory only, i.e. avoiding class field theory as a whole.

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I am no expert, but let me share an idea. For a number field $M$ let us denote by $C_M$ the idele class group of $M$. By class field theory, $K' \subset L'$ is the same as $N_{L'/K}(C_{L'})\subset N_{K'/K}(C_{K'})$, where $N$ stands for the norm map. Let us denote by $U$ the open subgroup $N_{L'/L}(C_{L'})$ of $C_L$, and by $j$ the natural injection of $C_K$ into $C_L$. Then, by the transfer theorem, the previous relation can be rewritten as $N_{L/K}(U)\subset j^{-1}(U)$, that is, $j(N_{L/K}(U))\subset U$. For certain open subgroups $U$ of $C_L$ this relation follows from ramification theory, and further local analysis might extend it to more general open subgroups.

EDIT: I think now that the conclusion $K' \subset L'$ fails when $L$ has two places $w$ and $w'$ above the same place $v$ of $K$ such that $L'/L$ is much more ramified at $w$ than $w'$. Indeed, viewing $U$ as a subgroup of the ideles $J_L$, this assumption implies that $U_w:=U\cap L_w^\times$ is much smaller than $U_{w'}:=U\cap L_{w'}^\times$. However, $j(N_{L/K}(U))\subset U$ would imply that $N_{L_{w'}/K_v}(U_{w'})\subset U_w$ which is false when $U_w$ is sufficiently small in terms of $U_{w'}$. Perhaps this argument can be rewritten in terms of the transfer map and ramification theory only, i.e. avoiding class field theory as a whole.

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I am no expert, but let me share an idea. For a number field $M$ let us denote by $C_M$ the idele class group of $M$. By class field theory, $K' \subset L'$ is the same as $N_{L'/K}(C_{L'})\subset N_{K'/K}(C_{K'})$, where $N$ stands for the norm map. Let us denote by $U$ the open subgroup $N_{L'/L}(C_{L'})$ of $C_L$, and by $j$ the natural injection of $C_K$ into $C_L$. Then, by the transfer theorem, the previous relation can be rewritten as $N_{L/K}(U)\subset j^{-1}(U)$, that is, $j(N_{L/K}(U))\subset U$. For the usual fundamental certain open subgroups $U$ of $C_L$ this relation follows from ramification theory, and further local analysis might extend it to more general open subgroups.

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