Noether's problem for abelian extensions of cyclic p-groups

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Let K be a field and G be a finite group. Let G act on the rational function field K(x(g):g∈ G) by K automorphisms defined by g· x(h)=x(gh) for any g,h∈ G. Denote by K(G) the fixed field K(x(g):g∈ G)ᵍ. Noether's problem then asks whether K(G) is rational (i.e., purely transcendental) over K. The first main result of this article is that K(G) is rational over K for a certain class of p-groups having an abelian subgoup of index p. The second main result is that K(G) is rational over K for any group of order p⁵ or p⁶ (p is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if char K≠ p then K contains a primitive pᵉ-th root of unity, where pᵉ is the exponent of G.)