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Hilbert class fielsfields and transfer

Let K/k$K/k$ be an extension of number fields and H_k$H_k$, H_K$H_K$ their respective Hilbert class fields. Is there a transfer map from Gal(H_k/k)$\text{Gal}(H_k/k)$ to Gal(H_K/K)$\text{Gal}(H_K/K)$?

Let K/k be an extension of number fields and H_k, H_K their respective Hilbert class fields. Is there a transfer map from Gal(H_k/k) to Gal(H_K/K)?

Let $K/k$ be an extension of number fields and $H_k$, $H_K$ their respective Hilbert class fields. Is there a transfer map from $\text{Gal}(H_k/k)$ to $\text{Gal}(H_K/K)$?

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Hilbert class fiels and transfer

Let K/k be an extension of number fields and H_k, H_K their respective Hilbert class fields. Is there a transfer map from Gal(H_k/k) to Gal(H_K/K)?