P-normal-extensible automorphism

Definition
Suppose $$p$$ is a prime number and $$P$$ is a p-group (i.e., a group where the order of every element is a power of $$p$$). An automorphism $$\sigma$$ of $$P$$ is termed a $$p$$-normal-extensible automorphism if, for any $$p$$-group $$Q$$ containing $$P$$ as a normal subgroup, there exists an automorphism $$\sigma'$$ of $$Q$$ whose restriction to $$P$$ equals $$\sigma$$.

Facts

 * Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible: Thus, for instance, all automorphisms of dihedral group:D8 are p-normal-extensible.