Distinguish the relations induced by all power-map digraphs and power-preserving bijections

Prove or disprove that there exist finite groups G and H such that the power-map digraphs of G and H are isomorphic for every exponent t, while no bijection between G and H simultaneously preserves all power maps.

Background

The paper defines G∼₂H to mean that there is a single bijection φ:G→H satisfying φ(gᵗ)=φ(g)ᵗ for every element g and every exponent t. It defines G∼₅H to mean that, for each t, the functional graphs of the power maps g↦gᵗ on G and H are isomorphic, with the isomorphism allowed to depend on t.

Theorem 1 establishes P₂=P₃=P₄≤P₅ for all finite groups, while Theorem 4 establishes P₂=P₃=P₄=P₅ for finite nilpotent groups. The unresolved issue is whether the equality P₂=P₅ extends from nilpotent groups to all finite groups, equivalently whether P₅<P₂ can occur.