Class-preserving automorphism group of finite p-group is p-group

Statement
Suppose $$P$$ is a finite $$p$$-group, i.e., a fact about::group of prime power order where the prime is $$p$$. Let $$\operatorname{Class}(P)$$ denote the group of fact about::class-preserving automorphisms of $$P$$, i.e., the automorphisms that send every element to within its conjugacy class. Then, $$\operatorname{Class}(P)$$ is also a $$p$$-group.

Facts used

 * 1) uses::Class-preserving implies stability automorphism of central series
 * 2) uses::Stability group of subnormal series of p-group is p-group

Proof
The proof follows directly from facts (1) and (2).