Noether's problem for central extensions of metacyclic $p$-groups

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 $g\cdot x(h)=x(gh)$forany$g,h\in G$\mathrm{.}Denoteby$K(G)$thefixedfield$K(x(g):g\in G)^G$\mathrm{.}Noethe{r}^{\mathrm{\prime }}sproblemthenaskswhether$K(G)$isrationalover$K$. In [M. Kang, Noether’s problem for metacyclic $p$-groups, Adv. Math. 203(2005), 554-567], Kang proves the rationality of $K(G)$ over $K$ if $G$ is any metacyclic $p$-group and $K$ is any field containing enough roots of unity. In this paper, we give a positive answer to the Noether’s problem for all central group extensions of the general metacyclic $p$-group, provided that $K$ is infinite and it contains sufficient roots of unity.