Characteristic subset of a group

Definition
A subset $$A$$ of a group $$G$$ is termed a characteristic subset if it satisfies the following equivalent conditions:


 * 1) For any automorphism $$\sigma$$ of $$G$$, $$\sigma(A) \subseteq A$$, i.e., $$\sigma(a) \in A$$ for all $$a \in A$$.
 * 2) For any automorphism $$\sigma$$ of $$G$$, $$\sigma(A) = A$$.
 * 3) For any automorphism $$\sigma$$ of $$G$$, the restriction of $$\sigma$$ to $$A$$ is a bijection from $$A$$ to itself.

Facts

 * Characteristic subset generates characteristic subgroup