Finite-extensible endomorphism implies trivial or automorphism

This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problem | View terms related to Extensible automorphisms problem

Statement

For a finite group, every extensible endomorphism is either the trivial map (i.e., it sends every element to the identity) or an automorphism.

Facts used

1. Every finite group is a subgroup of a finite simple group
2. Finite simple implies every endomorphism is trivial or an automorphism