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 $Aut_{c}^2(G)=Autcent(G)$.


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