The Hasse norm theorem mod squares

For a Galois extension K F of global fields, char F≠2, it is known that the Hasse norm theorem mod squares is equivalent to the existence of a local-global principle for the transfer ideal J_K/F of quadratic forms. When Gal(K/F)=(Z/2Z)^k, the Hasse norm theorem mod squares holds although the usual Hasse norm theorem may fail. In this paper we analyze the case when Gal(K/F)=Z/2Z⊕Z/2^kZ, k≥2. In particular, the Hasse norm theorem mod squares holds if and only if the Hasse norm theorem holds (Theorem I). A corollary shows that if a square in F is a local norm from K, then it is a global norm from K (Theorem II).