Fixed-class extensible automorphism

Definition with symbols
Let $$G$$ be a nilpotent group and let $$c$$ be the nilpotence class of $$G$$. An automorphism $$\sigma$$ of $$G$$ is said to be fixed-class extensible if, for any embedding $$G \le H$$ in a group $$H$$ of nilpotence class $$c$$, there exists an automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

Weaker properties

 * Finitely fixed-class extensible automorphism for a finite group
 * Fixed-class extensible endomorphism