# Extensible automorphism

A ${\displaystyle k}$ times extensible automorphism of a group is defined inductively as an automorphism that can be lifted to a ${\displaystyle k-1}$ times extensible automorphism for any embedding, where a 0 times extensible automorphism is simply any automorphism. An automorphism that is ${\displaystyle k}$ times extensible for all ${\displaystyle k}$ is termed an ${\displaystyle \omega }$ extensible automorphism. The ${\displaystyle k}$ extensible automorphisms of a group form a subgroup for every ${\displaystyle k}$.