Automorphisms of endomorphism semigroups of reflexive digraphs

A reflexive digraph is a pair (X,ρ), where X is an arbitrary set and ρ is a reflexive binary relation on X. Let End (X, ρ) be the semigroup of endomorphisms of (X, ρ). We determine the group of automorphisms of End(X,ρ) for: digraphs containing an edge not contained in a cycle, digraphs consisting of arbitrary unions of cycles such that cycles of length ≥ 2 are pairwise disjoint, and some circulant digraphs