Noether's problem for $p$-groups with an abelian subgroup of index $p$

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$-automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational over $K$. Let $p$ be an odd prime and let $G$ be a $p$-group of exponent $p^e$. Assume also that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that $K(G)$ is rational over $K$ for the following two types of groups: {\rm (1)} $G$ is a finite $p$-group with an abelian normal subgroup $H$ of index $p$, such that $H$ is a direct product of normal subgroups of $G$ of the type $C_{p^b}\times (C_p)^c$ for some $b,c:1\leq b,0\leq c$; {\rm (2)} $G$ is any group of order $p^5$ from the isoclinic families with numbers $1,2,3,4,8$ and 9.

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