Orthogonal units of the bifree double Burnside ring

The bifree double Burnside ring BΔ(G,G)BΔ(G,G) of a finite group G   has a natural anti-involution. We study the group B∘Δ(G,G) of orthogonal units in BΔ(G,G)BΔ(G,G). It is shown that this group is always finite and contains a subgroup isomorphic to B(G)×⋊Out(G)B(G)×⋊Out(G), where B(G)×B(G)× denotes the unit group of the Burnside ring of G   and Out(G)Out(G) denotes the outer automorphism group of G. Moreover it is shown that if G   is nilpotent then B∘Δ(G,G)≅B(G)×⋊Out(G). The results can be interpreted as positive answers to questions on equivalences of p-blocks of group algebras in the case that the block is the group algebra of a p-group.