  We study Frobenius extensions which are free-filtered by a totally ordered,
finitely generated abelian group, and their free-graded counterparts. First we
show that the Frobenius property passes up from a free-graded extension to a
free-filtered extension, then also from a free-filtered extension to the
extension of their Rees algebras. Our main theorem states that, under some
natural hypotheses, a free-filtered extension of algebras is Frobenius if and
only if the associated graded extension is Frobenius. In the final section we
apply this theorem to provide new examples and non-examples of Frobenius
extensions.
