Extensions of Hilbert class fields

Question

Let $K/F$ be a Galois extension of number fields. Is it true that the Hilbert class field $H_K$ of $K$ is an extension of the Hilbert class field $H_F$ of $F$ ?

If the class number of $F$ is $h_F > [K:F] h_K$, then we find a counterexample, but I don't know how to check this.

Answer 1

Yes, since the compositum $K\cdot H_F$ is an unramified Abelian extension of $K$, and so is contained in $H_K$.