Let be commutative rings with identity such that . A ring extension is called a -extension of rings if is a subring of for each pair of subrings of containing . In this paper, a characterization of integrally closed -extension of rings is given. The equivalence of -extension of rings and -extension of rings is established for an integrally closed extension of a local ring. Over a finite dimensional, integrally closed extension of local rings, the equivalence of -extensions of rings, FIP, and FCP is shown. Let be a subring of such that is invariant under action by , where is a subgroup of the automorphism group of . If is a -extension of rings, then is a -extension of rings under some conditions. Many such -invariant properties are also discussed.