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 $\mathbb{Q}$?

$\begingroup$ You should clarify if you want just $G$ to be realizable as Galois group, or to be realizable with quotient a given realization. This second case is called the embedding problem en.m.wikipedia.org/wiki/Embedding_problem $\endgroup$ – Xarles Oct 29 '18 at 7:32

$\begingroup$ I'm not asking for a realization of the quotient map. $\endgroup$ – user130124 Oct 29 '18 at 7:42
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 $p$.

1$\begingroup$ Have I got the wrong end of the stick? I thought that there weren't any groups that had been proved not to be the group of some Galois extension of the rationals. $\endgroup$ – Gerry Myerson Nov 16 '18 at 11:22

1$\begingroup$ I meant that $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ is not known to be the group of some Galois extension of the rationals not for lack of trying. $\endgroup$ – user130124 Nov 16 '18 at 16:36