Rationality problem for some meta-abelian groups

Let k be any field, G be a finite group. Theorem – Assume that (i) G contains an abelian normal subgroup H so that G/H is cyclic of order n, (ii) Z[ζn] is a unique factorization domain, and (iii) ζe∈k where e is the exponent of G, i.e. . If G→GL(V) is any finite-dimensional linear representation of G over k, then kG(V) is rational (= purely transcendental) over k.