Given a variety of finite monoids V, a subset of a monoid is a V-subset if its syntactic monoid belongs to V. A function between two monoids is V-preserving if it preserves V-subsets under preimages and it is hereditary V-preserving if it is W-preserving for every subvariety W of V. The aim of this paper is to study hereditary V-preserving functions when V is one of the following varieties of finite monoids: groups, p-groups, aperiodic monoids, commutative monoids and all monoids.