Let us denote by \(F\) a field with \(\text{char}(F) \neq 2\) and \(F^x/F^{x 2}\) finite. The aim of this paper is to determine all profinite 2-groups which occur as the Galois group \(G_F\) of the maximal 2-extension of a field \(F\), via the study of the Witt ring \(W(F)\) of quadratic forms over \(F\). By definition, \(W(F)\) has ''elementary type'' if it can be built from the indecomposable Witt ringsof finite, real-closed, and local fields, using the two operations of direct product (in the category of Witt rings) and group ring formation. When \(W(F)\) has elementary type, the authors give an algorithm effectively listing all possible pro-2-groups which could occur as \(G_F\). A crucial tool for the recursive description of the title is a ''realization theorem'' of abstract Witt rings in a paper by \textit{J. K. Arason, R. Elman} and the first author [J. Algebra 110, 449-467 (1987; Zbl 0629.10016)]. Tables are provided in the case \(|F^x/F^{x 2}| = 2,4,8\).