2-adic cofiltration of SO3(Q)

We prove that the group ${\mathrm{SO}}_3\left(Q\right)$ of rational rotations is the inverse limit of a family of finite solvable groups of order $2^{3k-2}\cdot 3$, whose $2$-Sylow subgroups have nilpotency class $2k-3$, exponent $2^{k-1}$, and Frattini subgroups coinciding with the commutator subgroups, and we give generators for these groups.