Prime set-extensible automorphism

Definition with symbols
Let $$\pi$$ be a set of primes. Let $$G$$ be a $$\pi$$-group, viz a finite group such that all prime divisors of its order are in $$\pi$$. An automorphism of $$G$$ is said to be $$\pi$$-extensible if it can be extended to an automorphism of $$H$$ for any $$\pi$$-group $$H$$ containing $$G$$.

Particular cases

 * Prime-extensible automorphism: A particular case when the prime set is a singleton set
 * Finite-extensible automorphism: A particular case when the prime set is the set of all primes