Suppose $P=L\ltimes U$ and $Q=M\ltimes V$ are $F$-stable parabolic subgroups of an algebraic group $G$, and $L$ and $M$ are $F$-stable Levi complements.
Let $S(L,M)=\{x\in G:L\cap\ ^xM\textrm{ contains a maximal torus of }G\}$.
If $x\in L^F\backslash S(L,M)^F/M^F$ (here $L^F$ is the $F$-fixed points of $L$, etc.) there is some functor $\operatorname{ad} x\colon\Lambda M\textrm{-mod}\to\Lambda\ ^xM\textrm{-mod}$ given by the action of $x$ by conjugation which shows up in the Mackey Formula.
How exactly does this functor work? I mean, if $N$ is a $\Lambda M$-module, what is the action of $^xM$ on $\operatorname{ad}x(N)$? If you have a double coset $L^FsM^F$ in $L^F\backslash S(L,M)^F/M^F$, an element of $^xM$ has form $L^FsM^FmM^Fs^{-1}L^F$, how does this act on $n\in\operatorname{ad}x(N)$?