Variety-extensible automorphism

Definition
Let $$\mathcal{V}$$ be a variety of algebras, and $$A$$ be an algebra in $$\mathcal{V}$$. An automorphism $$\sigma$$ of $$A$$ is termed extensible' over the variety $$\mathcal{V}$$ if, whenever $$A$$ is embedded as a subalgebra of an algebra $$B$$ of $$\mathcal{V}$$, there exists an automorphism $$\varphi$$ of $$B$$ such that the restriction of $$\varphi$$ to $$A$$ is $$\sigma$$.

Variety of sets
In the variety of sets, every automorphism is extensible. In other words, given any set and a subset, a permutation of the subset always extends to a permutation of the whole set.

The idea here is that adding more elements to a set does not destroy the inherent symmetry between the elements already there.

Variety of groups
In the variety of groups, every inner automorphism is extensible. The extensible automorphisms conjecture states that every extensible automorphism is inner.

Stronger properties

 * Weaker than::I-automorphism
 * Weaker than::Variety-infinity-extensible automorphism
 * Weaker than::Variety-pushforwardable automorphism
 * Weaker than::Variety-infinity-pushforwardable automorphism