P-extensible implies inner

Statement
Suppose $$p$$ is a prime number and $$P$$ is a p-group, i.e., a group in which the order of every element is a power of $$p$$. Suppose $$\sigma$$ is an automorphism of $$P$$ such that for any $$p$$-group $$Q$$ containing $$P$$, there is an automorphism $$\sigma'$$ of $$Q$$ whose restriction to $$P$$ equals $$\sigma$$. Then, $$\sigma$$ is an fact about::inner automorphism of $$P$$.