$\newcommand\Q{\mathbb Q}$Suppose $G$ is a transitive finite permutation subgroup of $S_{2n}$ having fixpoint-free involution in and outside its center. Suppose furthermore that I know that there is a (totally) imaginary Galois number field $K$ whose Galois group is isomorphic to $G$. So I also want $[K:\Q]=2n$. My question is the following:
If the complex conjugation is not in the center of $\operatorname{Gal}(K/\Q)$ then the maximal real subfield of $K$ is not Galois. But can I find another imaginary Galois number field $L$ with Galois group isomorphic to $G$ such that the complex conjugation is in the center of $G$? Vice versa, if the maximal real subfield of $\operatorname{Gal}(K/\Q)$ is Galois, can I find an imaginary number field $L$ with Galois group isomorphic to $G$ such that its maximal real subfield is not Galois?