Rings with finite maximal invariant subrings

We prove that if phi is an (anti-) automorphism of a ring R with finite orbits on R, or integral over the integers, and if R contains a finite maximal phi-invariant subring, then R must be finite. Special cases are when phi has finite order or is an involution. Two corollaries are that R must be finite when R contains only finitely many phi-invariant subrings or has both ascending and descending chain conditions on phi-invariant subrings. These generalize results in the literature for the special case when phi = id(R).