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Irreducible modules and normal subgroups of prime index

Let F be a field, G a finite group, H a normal subgroup of prime index p, and V an irreducible FH-module. If F is algebraically closed and of characteristic 0, the FG-module induced from V is either irreducible or a direct sum of p pairwise nonisomorphic irreducible modules. It is shown here that if F is not assumed algebraically closed and its characteristic is not 0, then there are not two but six possibilities for the structure of the induced module.