Let $(G,\cdot)$ be a finite group and $a,b \in G$. Defines another groups on $G$ under the binary operation $*_a$ and $*_b$ by $x*_ay = xay$ and $x*_b y = xby$, which are denoted by $(G,*_a), (G,*_b)$. Both are isomorphic to $(G,\cdot)$ by a mapping $x \mapsto xa^{-1}$ and $x \mapsto xb^{-1}$. Thus $(G,*_a)$ and $(G,*_b)$ are isomorphic under the map $xab^{-1}$. Define a simple graph on group $(G, \odot$) whose vertex set is $G$ and two distinct vertex $x,y$ are adjacent if $\exists z \in G$ such that $x, y \in \langle z \rangle$ and it is denoted by $P(G,\odot).$ Since $(G,*_a)$ and $(G,*_b)$ are isomorphic, so $P(G,*_a)$ and $P(G,*_b)$ are isomorphic graphs.
Can we any relate any isomorphism $\psi$ from $P(G,*_a)$ onto $P(G,*_b)$ to some automorphism $f$ of $P(G,*_a)$.
Thanks in advance for your kind help.