Solvable-extensible automorphism

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

Definition with symbols
Let $$G$$ be a solvable group and $$\sigma$$ an automorphism of $$G$$. Then $$\sigma$$ is said to be solvable-extensible if for any embedding of G in a solvable group $$H$$, there exists an automorphism $$\phi$$ of $$H$$ whose restriction to $$G$$ is $$\sigma$$.

Stronger properties

 * Extensible automorphism when the underlying group is solvable

Weaker properties

 * Nilpotent-extensible automorphism when the underlying group is nilpotent