Block relative trace characterizes diagonal projectivity

Suppose $B$ is relatively $\mathrm{\Delta }H$-projective. By Mackey its restriction to $\mathrm{\Delta }G$ is a summand of a sum induced from intersections $\mathrm{\Delta }G\cap (x,y)\mathrm{\Delta }H(x,y{)}^{-1}$. Each intersection has the form $\mathrm{\Delta }L$ with $L\le xH{x}^{-1}$: equality of the two components requires $xh{x}^{-1}=yh{y}^{-1}$. It is thus conjugate inside $\mathrm{\Delta }G$ into $\mathrm{\Delta }H$. Transitivity and conjugation of inducing witnesses make every term relatively $\mathrm{\Delta }H$-projective as a $\mathrm{\Delta }G$-module. The finite sum and its summand retain that property.

By Higman on this restricted module, ${\mathrm{i}\mathrm{d}}_B={\mathrm{Tr}}_{\mathrm{\Delta }H}^{\mathrm{\Delta }G}\alpha$. Evaluating at the central element $b$ gives $b={\sum }_{g\in G\mathrm{/}H}g\alpha (b)g^{-1}$. Put $a=\alpha (b)$; since $b$ is fixed by $\mathrm{\Delta }H$ and $\alpha$ is equivariant, $a\in B^H$.

Conversely let $b={\mathrm{Tr}}_H^G(a)$ with $a\in B^H$. Left multiplication $\alpha (v)=av$ commutes with the diagonal $H$ action, and its diagonal $G$ trace sends $v$ to $\sum ga{g}^{-1}v=bv=v$. Thus the restricted module is relatively $\mathrm{\Delta }H$-projective. By [F1] and the counit splitting, $B$ is a summand of induction to $G\times G$ of this restricted module. Induction transitivity now makes $B$ relatively $\mathrm{\Delta }H$-projective as a double module. For a block, minimizing over $p$-subgroups gives exactly [F2]. For $b=0$ both the zero module and trace witness $a=0$ satisfy the equivalence.