Every finite group is the Galois group of a field extension

How can I show that every finite group is the Galois group of an extension $K/F$ where $F$ is itself a finite extension of $\mathbb Q$?

I know the following:

1. Every finite group is contained in $S_p$ for a large enough prime $p$.

2. Every irreducible polynomial in $\mathbb Q[x]$ of degree $p$ having exactly $p-2$ real roots has a Galois group $S_p$ over $\mathbb Q$.

3. For any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.

Does this have something to do with the inverse Galois problem?

Well this is "a" inverse Galois problem in some sense but significantly easier than "the" inverse Galois problem.

Your three results basically solve the problem already.

Take your finite group $G$ and embed it in $S_p$ for some prime $p$ (via (1)). Take some irreducible polynomial $f$ over $\mathbb{Q}$ with exactly $p-2$ real roots (via (3)). Let $K$ be the splitting field of $f$ (over $\mathbb{Q}$) then $K$ has Galois group $S_p$ (via (2)). Set $F$ the fixed field of $G$ (considered as a subgroup of $S_p$). Then by the main theorem of Galois theory $G$ is the Galois group of $K/F$.