Automorphisms of soluble groups

Let R be a group of prime order r that acts on the r'-group G, let RG be the semidirect product of G with R, let 픽 be a field and V be a faithful completely reducible 픽[RG]-module. Trivially, CG(R) acts on CV(R). Let K be the kernel of this action. What can be said about K? This question is considered when G is soluble. It turns out that K is subnormal in G or r is a Fermat or half-Fermat prime. In the latter cases, the subnormal closure of K in G is described. Several applications to the theory of automorphisms of soluble groups are given