Get ready Part 26

Let $G$ be a group with identity $e$ and φ: $G \to G$ a function such that φ( $g_1$ )φ( $g_2$ )φ( $g_3$ ) =φ( $h_1$ )φ( $h_2$ )φ( $h_3$ ) whenever $g_1g_2g_3=e=h_1h_2h_3$ .Is there exists an element $a \in G$ such that ψ( $x$ ) = $a$ φ( $x$ ) is a homomorphism (i.e. ψ( $xy$ ) =ψ( $x$ )ψ( $y$ ) for all $x,y \in G$ )?