What's the maximal pro 2 Galois extension unramified outside 2, 3 and infinity over Q? I encounter a problem in my research: Let $L$ be the maximal pro-2 extension unramified outside $2, 3, \infty$ over $Q$, I hope I could know some information about the Galois group $Gal(L/Q)$. However, based on my poor knowledge of Galois theory, I have to admit that it's a little bit hard to me. Is there any people who could provide some information about that group? Any related information will be welcome. 
Thanks very much! 
 A: The answer to your question (in a slightly more general situation) is : Let $\mathbf Q_S/\mathbf Q$ be the maximal pro-2-extension unramified outside $S= ${$2, q, \infty$}, where $q$ is a prime $\equiv \pm 3$ mod $8$, and denote $G_S = Gal(\mathbf Q_S/\mathbf Q)$. Then $G_S$ can be described by generators and relations as follows : $G_S$ is the quotient of the free pro-2-group with generators $s_q, t_q, t_\infty$ modulo the normal subgroup generated by $t_q^{q-1}[t_q^{-1}, s_q^{-1}]$ and $t_\infty^2$. See the example 11.18 at the end of chap. 11 of [K]. How is this proved ? Because of your "poor knowledge of Galois theory" (*) , and consequently your even poorer knowledge of Galois cohomology, I can only give a very sketchy idea of the enormous machinery at work (see the references below).
Given a number field $K$, a rational prime $p$, and a finite set $S$ of primes of $K$ containing the set $S_p$ of primes of $K$ above $p$ as well as the set $S_{\infty}$ of infinite primes, the goal is the determination of $G_S = G_S (K)=Gal(K_S/K)$ where $K_S$ is the maximal pro-$p$-extension of $K$ unramified outside $S$. One must distinguish fundamentally between two cases because, by the definition of ramification, the following primes cannot ramify in a $p$-extension : the finite primes $P$ s.t. $N(P)\neq 1\pmod p$, the complex primes, and the real primes if $p$ is odd.
In the first case, $p$ is odd and $K$ is totally imaginary. In view of the "global-local" principle in ANT, one first describes by generators and relations the Galois group $G_v$ of the maximal pro-$p$-extensions of a local field $K_v$ ( = completion of $K$ at a prime $v\in S$). If $v\notin S_p$ (tame ramification), $G_v$ is described by a single simple relation. When  $v\in S_p$ (wild ramification) the problem is solved by two deep results : if $K_v$ does not contain a  primitive $p$-th root of unity, $G_v$ is pro-$p$-free (Shafarevitch), and if if $K_v$ contains such a root, $G_v$ is described by a single complicated relation (Demushkin). Back to the group $G_S$ : intuitively, its relations must include the local ones, but unfortunately, there are also truly global ones which, loosely speaking, are attached to the $\mathbf Z_p$-torsion subgroup of the abelianization of $G_S$. See [K], chap. 11, or [NSW], chap. 10 . 
The second case, which includes your question, is technically more complicated (as always when the prime $2$ is involved, think e.g. of the distinction between the class group and the narrow class group). See details  in [NSW], chap. 10, §6. Of course, your question can be entirely solved only thanks to the simplications brought by the fact that the base field is $\mathbf Q$ .
[K] H. Koch : "Galoissche Theorie der  $p$-Erxeiterungen"
[NSW] Neukirch-Schmidt-Wingberg : "Galois cohomology of number fields"  
(*) NB. But then, how did you get interested in this specific question ?
