Classify nonisomorphic groups equivalent under simultaneous power preservation

Determine all pairs of nonisomorphic finite groups (G,H) for which there exists a bijection φ:G→H satisfying φ(gᵗ)=φ(g)ᵗ for every g∈G and every exponent t.

Background

The relation ∼₂ is defined by the existence of a single bijection between two finite groups that commutes with every power map. Such a bijection need not be a group isomorphism, and the paper proves that this relation is strictly coarser than ordinary group isomorphism.

The authors note that groups of the same order and prime exponent provide examples: nonisomorphic groups of prime exponent p are equivalent under ∼₂. The problem asks for a complete determination of all such pairs, beyond the examples and products currently known.