I have been reading Brown's Book, Cohomology of groups and i cant quite see why a certain proposition is true, so he claims that
$Res_K^GInd_H^GM \cong \bigoplus_{g \in E} Ind_{K\cap gHg^{-1}}^{K} Res_{K\cap gHg^{-1}}^{gHg^{-1}}gM$, a K-isomorphism where $H$ and $K$ are subgroups of $G$ and $E$ is the set of representative classes for double cosets $KgH$, $Ind$ is the induction and $Res$ is the restriction of scalars.
I just put a picture here
Now i cant seem to prove this part b), the only thing i think its true is that $Ind_{H}^GM \cong \bigoplus_{g\in E} Ind_{K \cap gHg^{-1}}^K gM$ and then i guess we have to take $Res$ but i cant see why this would swich with $Ind$ and why those subgroups would appear. Any tips or advice are aprecciated. Thanks in advance.