Finite groups admitting mutually bijective dual Rota–Baxter operators

Determine for which finite groups G there exists a Rota–Baxter operator B such that both B and its dual operator B̃ are bijections.

Background

If the analogue of the algebraic spectral property fails, the paper observes that there may exist a nontrivial subgroup on which both a Rota–Baxter operator B and its dual B̃ act bijectively. This motivates a broader existence question concerning finite groups.

The paper identifies the Heisenberg group H₃ for p = 3 as the smallest nonabelian example by order and classifies the relevant operators on H₃. It also constructs examples on extraspecial groups of exponent p, but it does not determine all finite groups admitting such operators.