Studying class field theory, I have come across the following Proposition:
Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with $Gal(F/E)$ abelian. Then $h_E$ divides $h_K$.
In the proof, $H$ denotes the Hilbert class field of $E$ and it is derived that $Gal(HK/K)\cong Gal(H/E)$, which I was able to understand why it holds. Then the author says that this isomorphism also gives that $Gal(HK/K)$ is an unramified abelian extension of $K$.
My question is, why does the above hold and moreover, does this hold in a more general context? For example, if we have two number fields extension with isomorphic Galois groups and we know the ramification behavior in one of the extensions, what does this tell us about the ramification behavior in the other extension?
I am familiar with the notions of Decomposition group, Inertia group and Frobenius element, in case any of these are relevant to the answer.
Since $H/E$ is unramified and abelian, so is the translation HK/K. Ramification cannot become bigger by lifting the extension.
As for your second question: of course not. All quadratic extensions of the rational have isomorphic Galois groups.