Group property-conditionally normal-extensible automorphism

Definition
Suppose $$\alpha$$ is a group property, $$G$$ is a group satisfying $$\alpha$$, and $$\sigma$$ is an automorphism of $$G$$. We say that $$\sigma$$ is normal-extensible conditional to $$\alpha$$ if, for any group $$H$$ containing $$G$$ as a normal subgroup, there exists an automorphism $$\sigma'$$ of $$H$$ whose restriction to $$G$$ equals $$\sigma$$.

When $$\alpha$$ is the tautology, i.e., the property of being any group, the corresponding property is simply termed the property of being a normal-extensible automorphism.