# Is the following variant of Shafarevich's theorem known?

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 $$\mathbb{Q}$$?

• 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 – Xarles Oct 29 '18 at 7:32
• I'm not asking for a realization of the quotient map. – 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 $$p$$.
• 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. – user130124 Nov 16 '18 at 16:36