Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of
arbitrary characteristic, and let $H ring $k[V]$ of polynomial functions on $V$. We write $k[V]^G$ for the subalgebra
of $G$-invariant polynomials.

The extension of invariant rings
$k[V]^G\subset k[V]^H$ is studied using so called modules of covariants.
An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the
reflections on $V$ contained in $G$.
A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then
$G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module, then
$G$ is generated by $H$ and $W$. Furthermore, in that case the invariant ring $k[V]^{H\cap W}$
is free over $k[V]^W$ and is generated as an algebra by $H$-invariants and $W$-invariants.

This is joint work with Jianjun Chuai.