How is Harish-Chandra restriction compatible with Harish-Chandra series? Suppose $G$ is a connected reductive group over $\overline{\mathbb{F}}_p$, with Frobenius $F$. Let $(L_0,\Lambda_0)$ be a cuspidal pair with $L_0$ a Levi subgroup of a Levi subgroup $L$, and let ${^{\ast}{R^G_L}}$ denote the Harish-Chandra restriction functor. Supposedly, if $M$ is in the Harish-Chandra series for $G^F$ with respect to $(L_0,\Lambda_0)$, i.e., $M$ is a simple $G^F$-module and there is a surjection $R^G_{L_0}(\Lambda_0)\to M$, then any simple submodule of ${^\ast R^G_L(M)}$ is in the Harish-Chandra series for $L^F$ with respect to $(L_0,\Lambda_0)$.
How does one see this? If you have a simple submodule $N\to {^\ast R^G_L(M)}$, and a surjection $R^G_{L_0}(\Lambda_0)\to M$, how can you construct a surjection $R^L_{L_0}(\Lambda_0)\to N$?
Exploiting exactness, transitivity, and the adjunction yields maps like an injection $\Lambda_0\to {^\ast R^G_{L_0}(M)}$, a surjection $R^G_L(N)\to M$, and at least some nonzero morphism $R^L_{L_0}(\Lambda_0)\to {^\ast R^G_L(M)}$, but I don't see a way to piece these together to get the desired surjection.
 A: $\newcommand\Ind[3]{R_{#1}^{#2}(#3)}\newcommand\Res[3]{{^*R^{#1}_{#2}(#3)}}\DeclareMathOperator\Int{Int}\DeclareMathOperator\Norm{Norm}\DeclareMathOperator\SL{SL}$This is not literally true as stated.  For example, if $L = L_0 = A$ is a split maximal torus in, say, $G = \operatorname{SL}_2$, and if $\Lambda_0$ is any non-quadratic character of $L_0^F$ (i.e., $\Lambda_0^2 \ne 1$), then $M = \Ind{L_0}G{\Lambda_0}$ is irreducible, but $\Res G L M$ admits the Weyl conjugate $(L_0, \Lambda_0^{-1})$ of $(L_0, \Lambda_0)$ as quotient.
Based on the notation, I think you are using Digne and Michel - Representations of finite groups of Lie type.  I will cite from there.
What is true is that, in your setting, there is some $g \in G^F$ such that $N$ lies in the Harish-Chandra series of $(\Int(g)L_0, \Lambda_0 \circ \Int(g)^{-1})$.  I argue as in my comment.  Namely, since all representations are unitary, two representations admit a non-$0$ intertwining map (either way) if and only if they share a composition factor, and all (simple) composition factors are both submodules and quotient modules (Lemma 5.3.6).  Because of the symmetry, I will just say that two representations intertwine if there is a non-$0$ intertwining map between them (in either direction).  Thus $M$ is a submodule of $\Ind{L_0}G{\Lambda_0}$, so that $N$ is a submodule of $\Res G L{\Ind{L_0}G{\Lambda_0}}$.  On the other hand, by Theorem 5.3.7, there is some cuspidal pair $(L_0', \Lambda_0')$ in $L$ such that $N$ is a quotient, hence a submodule, of $\Ind{L_0'}L{\Lambda_0'}$.  Thus $\Res G L{\Ind{L_0}G{\Lambda_0}}$ and $\Ind{L_0'}L{\Lambda_0'}$ share a simple submodule, hence intertwine; so, by Frobenius reciprocity (Proposition 5.1.3), using transitivity (Proposition 5.1.4(i)), there is also a non-$0$ intertwining map $\Ind{L_0'}G{\Lambda_0'} = \Ind L G{\Ind{L_0'}L{\Lambda_0'}} \to \Ind{L_0}G{\Lambda_0}$.  Now the desired fact follows from another application of Theorem 5.3.7.
(I have edited the previous paragraph to correct the typo ($N$ for $\Lambda_0'$) @AdelaideDokras pointed out.)
Incidentally, in the $p$-adic setting with which I am more familiar, the cuspidal pair to whose Harish-Chandra series a representation belongs is called its cuspidal support; Harish-Chandra induction and restriction are usually called parabolic induction and Jacquet restriction, respectively; and both depend, at the level of representations (rather than of the Grothendieck group), on a choice of parabolic, not just of Levi.  See Casselman's $p$-adic notes for an excellent reference.  In that setting, we see that, though the individual representations are different, all choices of parabolic yield the same semisimplified representation (i.e., have the same composition factors, counted with multiplicity) (Theorem 6.3.11).  In that setting, one has to replace my free-wheeling disregard of the difference between simple submodules and quotient submodules with the convenient fact that one can 'move' any composition factor of the parabolic induction down to the bottom of the composition series (i.e., realise it as a submodule), possibly by choosing a different parabolic with the same Levi component (Corollary 6.3.7).  (The irreducibility result I quote above is Theorem 6.6.1.)  Then the corrected result above is Theorem 6.3.6.
