Is there a Mackey Formula for the composite of restriction first, then induction? Every instance of the Mackey restriction formula I've ever encountered gives a way of decomposing $\operatorname{Res}^G_K\operatorname{Ind}_H^G W$ as a sum of smaller representations parametrized by double cosets. Likewise, for Harish-Chandra induction, the Mackey formula applies to the composite $^\ast R^G_L\circ R^G_M$, where the Harish-Chandra induction functor $R^G_M$ is applied first, and then the Harish-Chandra restriction functor $^\ast R_L^G$. (Here I'm just denoting $L$ and $M$ to be some Levi subgroups of an algebraic group $G$.) The similarity here is that the induction functor always is applied first.
Is there an analogous formula for the case where one restricts first, then inducts? Say for the composite $\operatorname{Ind}_H^K\operatorname{Res}^G_H$ or $R^L_M\circ {^\ast}R^G_M$? I've never seen one mentioned.
 A: There's an important difference between the two cases. Let me start by putting things in a more general context.
If $G$ and $H$ are two groups, define a $(G,H)$-biset to be a set $X$ together with commuting actions of $G$ on the left and of $H$ on the right. Equivalently $X$ is a set with a right action of $G\times H$ by $x\cdot(g,h)=g^{-1}xh$. Then for representations over a coefficient field $k$, $X$ gives rise to a functor from (right) $kG$-modules to $kH$-modules as follows: 
Let $kX$ be the vector space with basis $X$. Then $kX$ is a $kG$-$kH$-bimodule, and we can take the functor
$$F_X=-\otimes_{kG}kX: \text{mod-}kG\to\text{mod-}kH.$$
Many familiar functors are of this form. For example, if $H$ is a subgroup of $G$ and $X=G$ with $G$ and $H$ acting by left and right multiplication, then $F_X$ is just restriction. If $G$ is a subgroup of $H$ and $X=H$, then $F_X$ is just induction.
Compositions of functors of this kind are also of the same kind. If $X$ is a $(G,H)$-biset and $Y$ is an $(H,K)$-biset, then
$$F_Y\circ F_X=F_{X\times_HY},$$
where $X\times_HY$ is the set of equivalence classes in $X\times Y$ for the equivalence relation given by $(xh,y)\sim(x,hy)$ for $x\in X$, $y\in Y$, $h\in H$.
If a $(G,H)$-biset $X$ has more than one orbit for the action of $G\times H$, then it is the disjoint union of $(G,H)$-bisets $X_1,\dots,X_n$, and then $kX=kX_1\oplus\dots\oplus kX_n$, and so we get a decomposition of the functor $F_X$ in terms of the orbits.
If $H$ and $K$ are subgroups of $G$, so
$$\text{Res}^G_K\text{Ind}^G_H=F_{G\times_GG},$$
then the $(H,K)$-biset $G\times_GG$ is just $G$ with $H$ and $K$ acting by left and right multiplication, and this decomposes as a biset into the disjoint union of double cosets, and this gives the Mackey decomposition formula.
However, if $H$ is a subgroup of $G$ and $K$, then
$$\text{Ind}^K_H\text{Res}^G_H=F_{G\times_HK},$$
and the action of $G\times K$ on $G\times_HK$ is transitive, so here there is no non-trivial decomposition in terms of orbits.
