Normal-extensible automorphism

Symbol-free definition
An automorphism of a group is termed normal-extensible if, for any embedding of the group as a normal subgroup of another group, the automorphism can be extended to an automorphism of the bigger group.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed normal-extensible if, for any embedding of $$G$$ as a normal subgroup of another group $$H$$ there is an automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

Formalisms
The property of normal-extensibility arises by applying the qualified extensibility operator with the qualifying property being normality and the the automorphism property being the tautology (that is, the property of being any automorphism).

Stronger properties

 * Weaker than::Extensible automorphism: . Also related:
 * Weaker than::Inner automorphism
 * Weaker than::Subnormal-extensible automorphism
 * Iteratively normal-extensible automorphism viz $$\alpha$$-normal extensible automorphism for an ordinal $$\alpha$$ that is at least 1
 * Normal series-extensible automorphism

Weaker properties

 * Stronger than::Characteristic-extensible automorphism
 * Stronger than::Central factor-extensible automorphism

Incomparable properties

 * Normal automorphism:

Related group properties

 * Group in which every automorphism is normal-extensible
 * Group in which every normal-extensible automorphism is inner

Facts

 * Centerless and maximal in automorphism group implies every automorphism is normal-extensible
 * Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
 * NRPC conjecture implies normal-extensible equals characteristic-extensible

More facts about characterizing normal-extensible automorphisms can be found at the normal-extensible automorphisms problem page.

Metaproperties
The collection of normal-extensible automorphisms of a group form a subgroup of the automorphism group. This follows from the general fact that the qualified extensibility operator is a group-closure-preserving automorphism property operator.