  Let $G$ be a finite group and $k$ be a field. 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 $g,h\in G$. Noether's problem asks whether the fixed field
$k(G)=k(x_g:g\in G)^G$ is rational (i.e. purely transcendental) over $k$. We
will prove that, if $G$ is a group of order $2^n$ ($n\ge 4$) and of exponent
$2^e$ such that (i) $e\ge n-2$ and (ii) $\zeta_{2^{e-1}} \in k$, then $k(G)$ is
$k$-rational.13A50,14E08,14M20,12F12
