Nilpotent-extensible automorphism

Symbol-free definition
An automorphism of a nilpotent group is said to be nilpotent-extensible if it can be extended to an automorphism for every embedding of the given nilpotent group into a nilpotent group.

Definition with symbols
Let $$G$$ be a nilpotent group and $$\sigma$$ an automorphism of $$G$$. Then we say that $$\sigma$$ is nilpotent-extensible if for any embedding of $$G$$ in a nilpotent group $$H$$, there is an automorphism $$\phi$$ of $$H$$ such that the restriction of $$\phi$$ to $$G$$ is $$\sigma$$.

Stronger properties

 * Extensible automorphism when the underlying group is nilpotent
 * Solvable-extensible automorphism when the underlying group is nilpotent