Here are some musings I've had about the Inverse Galois Problem, which I'm curious as to what thoughts have been given to them. I know only a little of Galois theory so forgive me if these are just plain wrong, or evident results.
Let $G = \textbf{Gal}(\bar{\mathbb{Q}} / \mathbb{Q})$, and assume that $G$ has a nontrivial finite subgroup $H$ (do we even know if such an $H$ exists?).
Is $H$ the Galois group of a polynomial?
The converse question is also interesting.
If $K$ is the Galois group over $\mathbb{Q}$ of a polynomial, is it a subgroup of $G$?
Furthermore, and I'm not sure how to properly phrase this, if $g$ is an element of $K$, can it be extended to an element of $G$? In the sense that if $f(x)$ is a polynomial such that $K$ is its Galois group, and $\hat{g}$ is the extended element, $\hat{g}$ acts on the roots of $f(x)$ just like $g$.
If the first question is true, then the Inverse Galois Problem could be reformulated as showing that every finite group is a subgroup of $G$. Does any of these questions make any sense? I guess this approach won't really do much since we don't really know anything about the structure of $G$.