A Galois extension is considered universally consistent with the period q if for any problem of embedding of this extension with an abelian kernel of the period q the consistency condition holds. Let K be a universally consistent extension of the period 2n + 1 of an algebraic number field k, such that 2 completely splits in K, and let (K/k, φ) be an embedding problem with the cyclic kernel of order 2. It is proven that (under some group-theoretical restrictions) there exists a solution of this embedding problem that is universally consistent with the period 2n.