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, and let H < G be a subgroup. The extension of invariant rings k[V](G) subset of 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. Furthermore, the invariant ring k[V]H boolean AND W is free over k[V](W) and is generated as an algebra by H-invariants and W-invariants.