Number of orbits for finite group action on finite vector space equals number of orbits on dual vector space

Statement
Suppose $$G$$ is a finite group, $$F$$ is a finite field, and $$V$$ is a finite-dimensional vector space over $$F$$. Suppose we have a linear representation of $$G$$ on $$V$$, i.e., a homomorphism $$G \to GL(V)$$. The contragredient representation is a linear representation of $$G$$ on $$V^*$$, i.e., a homomorphism $$G \to GL(V^*)$$.

The claim is that the number of orbits of $$V$$ under the original $$G$$-representation equals the number of orbits of $$V^*$$ under the contragredient representation.

Journal references

 * , Lemma 4.1