Generic initial ideals of modular polynomial invariants

We study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all the cases where an explicit generating set is known, we calculate the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also clarify gin for the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all the situations we consider, there is a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.