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