Extensible endomorphism

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

Related terms
An extensible automorphism is an automorphism that can always be extended to an automorphism for any bigger group. Every extensible automorphism is an extensible endomorphism and also an automorphism. However, there is no a priori reason that every extensible endomorphism that is also an automorphism should be an extensible automorphism.

A finite-extensible endomorphism is an endomorphism of a finite group that can always be extended to an endomorphism for any finite group containing it.

Facts

 * Finite-extensible endomorphism implies trivial or automorphism