Noether's problem for p-groups with a cyclic subgroup of index p2

Let K be any field and G be a finite group. Let G act on the rational function field K(xg:g∈G) by K-automorphisms defined by g⋅xh=xgh for any g,h∈G. Noether's problem asks whether the fixed field K(G)=K(xg:g∈G)G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order pn (n⩾3) containing a cyclic subgroup of index p2 and K is any field containing a primitive pn−2-th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p3 and K is a field containing a primitive p-th root of unity, then K(G) is rational.