A generalizaton of nilpotent groups

Question

‎‎‎‎‎‎‎Let ‎$‎G‎$ ‎be a‎ ‎group ‎and ‎‎$‎‎\alpha\in Aut(G)$ ‎be a‎ ‎fixed ‎automorphism ‎of ‎‎$‎G‎$‎. An ‎$‎‎\alpha$-commutator ‎of ‎elements ‎‎$‎‎x, y\in G$ ‎is ‎‎$‎‎[x, y]_{\alpha}= x^{-1}y^{-1}xy^{\alpha}$. ‎The ‎‎$‎‎\alpha$-center ‎subgroup ‎of ‎‎$‎G‎$‎, denoted by ‎$‎‎Z^{\alpha}(G)$ ‎is ‎defined ‎as ‎‎$‎‎Z^{\alpha}(G)= \{x\in G : [y, x]_{\alpha}= 1, ‎\forall ‎y\in ‎G‎ \}‎‎$. ‎ ‎If ‎$‎‎N$ ‎is a‎ ‎normal ‎subgroup ‎of ‎‎$‎G‎$ ‎which ‎is ‎invariant ‎under ‎‎$‎‎\alpha$ ‎and‎ ‎‎$\bar{\alpha}‎$ is an automorphism of quotient group‎‎‎‎‎ ‎‎‎‎‎$‎G/N$ ‎such that ‎send ‎an ‎element ‎‎$‎‎gN$ ‎to ‎‎$‎‎g^{\alpha}N$‎, then ‎the ‎following normal ‎series‎ $$ ‎\{ ‎1\}= ‎G_{0}‎\unlhd ‎G_{1}‎\unlhd ‎\dots‎ ‎\unlhd ‎G_{n}= ‎G‎, ‎‎$$ ‎‎is called a central ‎$‎‎\alpha$‎-series whenever ‎$‎‎G_{i}^{\alpha}= G_{i}$ ‎and ‎‎$‎‎G_{i+1}/G_{i}‎\leq Z^{\bar{\alpha}}(G/G_{i})‎$‎, for ‎$‎‎0‎\leq i‎\leq n-1‎‎$‎.‎‎ ‎An ‎‎$‎‎\alpha$-nilpotent ‎group ‎is a‎ ‎group ‎which ‎possesses ‎at ‎least a‎ ‎central ‎‎$‎‎‎\alpha‎$‎-series. It is to see that f $\alpha$ is an inner automorphism of a nilpotent group $G$, then $G$ is an $\alpha$-nilpotent group. My Question is: Is there any non-inner automorphim $\alpha$ of a finite non-abelian $p$-group $P$, such that $P$ is $\alpha$-nilpotent?