Let $G$ be finite group and $k$ a field. Inverse Galois theory asks if there is a galois extension $L/k$ such that $Gal(L/k) \simeq G$. Lets assume $k=\mathbb{Q}$ and let $\mathcal{h}_L$ denote the the class number of $L/k$.
If there are non isomorphic realizations of $G$ via $L_1/k, L_2/k$, what can one say about the class numbers $\mathcal{h}_{L_1}$, $\mathcal{h}_{L_2}$?
If it is possible that they dont coincide what is the possible amount these numbers differ by?