Endomorphisms of undirected modifications of directed graphs

If z is an automorphism (or an endomorphism) of H, then it is also an automorphism (or an endomorphism) of %H and of %H. Hence the automorphism group Aut H is a subgroup of Aut 9?H and of Aut %H. Is there any other relation between the groups Aut H, Aut CBH and Aut %H? More in detail, which spans of groups, see Fig. 1, where m,, m2 are monomorphisms, can be realized by directed graphs H (in the sense that there exist isomorphisms qO of G,, onto Aut H, q~, of G, onto Aut %JH and q2 of G2 onto Aut %H such that the diagram in Fig. 2-where i,, i2 are the inclusion maps-commutes)? In the present paper, we prove the following.