Finite-extensible endomorphism

Definition
Let $$G$$ be a finite group and $$\alpha$$ be an endomorphism of $$G$$. We say that $$\alpha$$ is a finite-extensible endomorphism of $$G$$ if, for any group $$H$$ containing $$G$$, there exists an endomorphism $$\alpha'$$ of $$H$$ such that the restriction of $$\alpha'$$ to $$G$$ equals $$\alpha$$.

It turns out that any finite-extensible endomorphism must be an automorphism.

Facts

 * Finite-extensible endomorphism implies trivial or automorphism: This in turn follows from the fact that every finite group is a subgroup of a finite simple group.