Kaplansky's radical and a recursive description of pro-2 Galois groups

In this note we study a modified version of the ``Elementary Type Conjecture'' for pro-$2$ Galois groups and its connection with the Kaplansky's radical. To be more precise, for a field $F$ of characteristic $\ne 2$ let $F(2)$ be its quadratic closure and denote by $G_2(F)$ the corresponding Galois group. We state a condition, involving the Kaplansky's radical of $F$, which implies that $G_2(F)$ can be obtained from some suitable closed subgroups using free pro-$2$ products and semi-direct group extension operations a finite number of times.