TITLE: Galois Group of the Hilbert Class Field
QUESTION [21 upvotes]: Let $K/\mathbb{Q}$ be a number field with Galois group G and let $L/K$ be the Hilbert class field of $K.$ It is easy to show that $L$ is Galois over $\mathbb{Q}$ and I am interested in knowing this Galois group. 
We know that the class group of $K$ is isomorphic to the Galois group $L/K$ so I would assume that $\operatorname{Gal}(L/\mathbb{Q})\cong cl K \rtimes G.$ However this (somewhat unrelated) MO post (esp. Kevin Buzzards comment on Felipe Voloch's answer) seems to indicate otherwise. Can somebody shine light on the situation?
Thanks.

REPLY [3 votes]: C. S. Herz (Construction of Class Fields,  Seminar on complex multiplication (Chowla et al. eds.) (1957), Lecture Notes Math. 21, Springer Verlag) has claimed  that the answer to your question is positive. The first counterexample was provided by Wyman (Hilbert class fields and group extensions,  Scripta math. 29 (1973), 141-149). Simpler proofs and more examples were given later by

R. Gold, Hilbert class fields and split extensions, 
Ill. J. Math. 21 (1977), 66-69
G. Cornell, M. I. Rosen, A note on the splitting of the Hilbert class field, 
J. Number Theory 28 (1988), 152-158
R. Bond, On the splitting of the Hilbert class field, 
J. Number Theory 42 (1992), 349-360