The Noether map II

Let $\rho: G\hookrightarrow \mathrm{GL}(n, \mathbb{F})$ be a faithful representation of a finite group $G$. In this paper we proceed with the study of the image of the associated Noether map

$\eta_G^G: \mathbb{F}[V(G)]^G \rightarrow\mathbb{F}[V]^G.$

In our 2005 paper it has been shown that the Noether map is surjective if $V$ is a projective $\mathbb{F} G$-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for $p$-groups (where $p$ is the characteristic of the ground field $\mathbb{F}$) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of $V$.