Modules of covariants in modular invariant theory

Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\subset k[V]^H$ is studied using modules of covariants. An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections 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 $W$, and the invariant ring $k[V]^{H\cap W}$ is free over $k[V]^W$ and generated as an algebra by $H$-invariants and $W$-invariants.