The deviation conjectures for finite groups and subgroups

Let $G$ be a finite group and let $H<G$. Define the deviation $\Delta(G,H)$ as the smallest natural number $\Delta$ such that $\Delta\cdot[H\backslash H]$ lies in the image of

$$\operatorname{Res}\,\operatorname{Ind}:\mathcal{B}(H)\to\mathcal{B}(H).$$

Deviation conjectures. With $\Delta=\Delta(G,H)$, one has $\Delta=1$ if and only if $H=G$, the index $[G:H]$ divides $\Delta$, and $\Delta$ divides $|G|$. These conjectures propose arithmetic constraints on the deviation, which measures both the failure of normality and the relative size of $H$ in $G$.