AbstractA 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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)