Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL\_2(\mathbb{F}\_p)$ for $p\geq 5$, or the monster group) and let $G$ be an extension of $Q$ by a cyclic group. Then is it possible to see that $G$ is realizable as a Galois group over $\mathbb{Q}$?



Is the following variant of Shafarevich's theorem known?
It has occurred to me not too long after posting this question that the answer is obviously no, for instance the groups $\text{PSL}\_2(\mathbb{Z}/p\mathbb{Z})$ are known to be Galois over $\mathbb{Q}$ but the groups $\text{SL}\_2(\mathbb{Z}/p\mathbb{Z})$ are not known to be Galois over $\mathbb{Q}$ for large $p$.


