Near automorphisms of powers of a path

Let $G$ be a connected graph with vertex set $V\left(G\right)$, and $f$ a permutation on $V\left(G\right)$. Define ${\delta }_f(x,y)=|d(x,y)-d(f\left(x\right),f\left(y\right))|$ and ${\delta }_f\left(G\right)=\sum {\delta }_f(x,y)$, where the sum is taken over all unordered pairs $x,y$ of distinct vertices of $G$. Denote by $\pi \left(G\right)$ the smallest positive value of ${\delta }_f\left(G\right)$ among all permutations on $V\left(G\right)$. A permutation $f$ on $V\left(G\right)$ is called a near automorphism of $G$ if ${\delta }_f\left(G\right)=\pi \left(G\right)$. The $k$th power of $G$, written as $G^k$, also has $V\left(G\right)$ as vertex set, but $x$ and $y$ are adjacent in $G^k$ whenever $d(x,y)\le k$ in $G$. The near automorphisms of the path $P_n$ and its square $P_n^2$ were described in Aitken (1999) and Wong et al. (2023), respectively. In this paper, we give the value of $\pi \left(P_n^k\right)$, and determine the near automorphisms of $P_n^k$ when $3\le k\le n-2$.