TITLE: Galois groups and prescribed ramification
QUESTION [7 upvotes]: What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can be realized over $\mathbb{Q}$ with no ramification outside a given (finite) set of primes?
I am thus interested in results of two kinds:
Realization of specific groups.
Examples of groups which are not realizable in this manner, and restrictions on groups which are.

REPLY [8 votes]: Concerning the question of Pablo that follows Joël's answer:
If $k$ is an algebraically closed field, then the situation is completely understood, thanks to work of Grothendieck (in characteristic 0) completed by the proof of Abhyankar's conjecture by Raynaud and Harbater. 
Precisely, let $C/k$ be a smooth affine curve, complement of $r\geq1$ points in a projective smooth connected curve of genus $g$, over the alg. closed field $k$.
Let $F=k(C)$ be the field of rational functions on $C$.
Extensions of $C$ that are unramified on $C$ correspond to connected 
étale covers of $C$.
If $\mathop{\rm char}(k)=0$, then Grothendieck showed (this is the main result of SGA 1) that a finite group $G$ is the Galois group of a connected étale cover of $C$ if and only if it is generated by $2g+r-1$ elements. More precisely, the Galois group of the maximal extension of $k(C)$ unramified on $C$ is a free profinite group on $2g+r-1$ elements. It is worth observing the fundamental group of a compact connected Riemann surface of genus $g$ deprived from $r\geq1$ points is  a free group on $2g+r-1$ generators. (The case $r=0$ is also treated by Grothendieck, $G$ then needs to be generated by $2g$ elements $a_1,b_1,\dots,a_g,b_g$ satisfying the relation $(a_1,b_1)(a_2,b_2)\dots(a_g,b_g)=1$.)
If $\mathop{\rm char}(k)=p>0$, then $C$ has « more » étale covers. For example,
if $C=\mathbf A^1_k$, then $f\colon C\to C$ given by $f(t)=t-t^p$ is a connected étale cover of degree $p$. Raynaud (when $C=\mathbf A^1_k$) and Harbater (in general) showed that a finite group $G$ is the Galois group of a connected étale cover of $C$ if and only if the quotient of $G$ by the (normal) subgroup generated by its $p$-Sylow subgroups is generated by $2g+r-1$ elements.
Specific examples had been exhibited by many mathematicians, such as Abhyankar itself (he has a long series of papers in this style) and Nori (when $G$ is a finite group of Lie type).