# On $n$th class-preserving automorphism of finite $p$-group

Let $$G$$ be a finite non-abelian $$p$$-group, where $$p$$ is a prime. An automorphism $$\alpha$$ of $$G$$ is called an $$n$$th class-preserving if for each $$x\in G$$, there exists an element $$g_x\in \gamma_n(G)$$ such that $$\alpha(x)=g_x^{-1}xg_x$$, where $$\gamma_n(G)$$ denotes the $$n$$th term of the lower central series of $$G$$. An automorphism $$\alpha$$ of $$G$$ is called a central automorphism if $$x^{-1}\alpha(x)\in Z(G)$$ for all $$x\in G$$. Let $$Aut_{c}^n(G)$$ and $$Autcent(G)$$ respectively denote the group of all $$n$$th class-preserving and central automorphisms of $$G$$.

My question is the following: Give some examples of finite non-abelian $$p$$-group $$G$$ of nilpotency class 3 such that $$Aut_{c}^2(G)=Autcent(G)$$.