Two examples in the Galois theory of free groups

AbstractLet F be a free group, and let H be a subgroup of F.The ‘Galois monoid’ EndH(F) consists of all endomorphisms of F which fix every element of H; the ‘Galois group’ AutH(F) consists of all automorphisms of F which fix every element of H. The End(F)-closure and the Aut(F)-closure of H are the fixed subgroups, Fix(EndH(F)) and Fix(AutH(F)), respectively.Martino and Ventura considered examples whereFix(AutH(F))≠Fix(EndH(F))=H. We obtain, for two of their examples, explicit descriptions of EndH(F), AutH(F), and Fix(AutH(F)), and, hence, give much simpler verifications that Fix(AutH(F))≠Fix(EndH(F)), in these cases