Every group embeds malnormally into a simple complete co-Hopfian group, so non-trivial endomorphisms extend to the larger group if and only if they are inner automorphisms.
70, Kluwer Academic Publishers Group, Dordrecht, 1991, Translated from the 1989 Russian original by Yu
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An improved characterisation of inner automorphisms of groups
Every group embeds malnormally into a simple complete co-Hopfian group, so non-trivial endomorphisms extend to the larger group if and only if they are inner automorphisms.