Naturality in real Heegaard Floer theory

In a previous paper we defined real Heegaard Floer homology, an invariant of three-manifolds equipped with involutions. Here we prove that real Heegaard Floer homology is natural, and that it admits an action of the equivariant mapping class group. We also establish naturality of real link Floer homology and real sutured Floer homology, and give a definition of involutive real Heegaard Floer homology.