Orthogonal units of the bifree double Burnside ring

The bifree double Burnside ring $B^{\mathrm{\Delta }}(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B_{\circ }^{\mathrm{\Delta }}(G,G)$ of orthogonal units in $B^{\mathrm{\Delta }}(G,G)$. It is shown that this group is always finite and contains a subgroup isomorphic to $B(G{)}^{\times }⋊\text{Out}(G)$, where $B(G{)}^{\times }$ denotes the unit group of the Burnside ring of $G$ and $\text{Out}(G)$ denotes the outer automorphism group of $G$. Moreover it is shown that if $G$ is nilpotent then $B_{\circ }^{\mathrm{\Delta }}(G,G)\cong B(G{)}^{\times }⋊\text{Out}(G)$. 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.