Parabolic Induction in Stages $\DeclareMathOperator{\Ind}{Ind}$Let $G$ be a connected, reductive group over a local field $k$.  
Let $P_{\ast} = M_{\ast}N_{\ast} \subseteq P = MN$ be parabolic subgroups, standard with respect to a given minimal parabolic, with standard Levi subgroups $M_{\ast} \subseteq M$.  
Then $M_{\ast}(N_{\ast} \cap M)$ is a parabolic subgroup of $M$ with unipotent radical $N_{\ast} \cap M$.  If $(\pi,V)$ is a representation of $M_{\ast}$, then we can regard $\pi$ as a representation of $P_{\ast}$ by making it trivial on $N_{\ast}$, and form the induced representation $\Ind_{P_{\ast}}^G \pi$.  On the other hand, we can regard $\pi$ as a representation of $M_{\ast}(N_{\ast} \cap M)$ by making it trivial on $N_{\ast} \cap M$, and form the induced representation
$$\sigma = \Ind_{M_{\ast}(N_{\ast} \cap M)}^M \pi$$
Then we can extend $\sigma$ to $P$ by making it trivial on $N$, and form
$$\Ind_P^G \sigma$$
I expect that we should be able to identify the representations
$$\Ind_P^G\Ind_{M_{\ast}(N_{\ast} \cap M)}^M \pi = \Ind_{P_{\ast}}^G \pi$$
This initially appears to be an application of the transitivity of induction, but this principle cannot be immediately implied.  Are these two representations of $G$ the same?  What if we do these by normalized induction?  (Taking into account the modulus characters)
 A: $\DeclareMathOperator{\Hom}{Hom}$I think that this follows from a version of Frobenius reciprocity: if $(\tau,W)$ is a representation of $G$, and $W_N = W/W(N)$ is the Jacquet module of $P$ (where $W(N)$ is the linear span of $w - \tau(n)w : w \in W, n \in N$), then we have
$$\Hom_G(W, \Ind_P^G\pi) = \Hom_M(W_N,\pi)$$
Here $W_N$ inherits the structure of a representation of $M$.  We can further restrict this to a representation of the parabolic subgroup $M_{\ast}(N_{\ast} \cap M)$ of $M$ and take the Jacquet module again.
Lemma: The Jacquet module $W_N/[W_N(N_{\ast} \cap M)]$ coincides with $W_{N_{\ast}}$ as a representation of $M_{\ast}$.
Proof: By the third isomorphism theorem, this comes down to showing that the subspace $W_N(N_{\ast} \cap M)$ of $W_N = W/W(N)$ is equal to
$$W_N(N_{\ast} \cap M) = \frac{W(N_{\ast})}{W(N)}$$
It is clear that the left hand side is contained in the right.  Now consider a generator $w - \tau(n)w + W(N)$ of the right hand side, for some $w \in W$ and $n \in N_{\ast}$.  Since $N_{\ast}$ is the semidirect product of $N$ by $N_{\ast} \cap M$, we can write $n = n_1n_2$ for $n_1 \in N_{\ast} \cap M$ and $n_2 \in N$.  Then
$$w - \tau(n)w + W(N) = w - \tau(n_2)w + \tau(n_2)w - \tau(n)w+W(N) = \tau(n_2)w - \tau(n)w + W(N)$$
with $\tau(n_2)w - \tau(n)w = \tau(n_2)w - \tau(n_1)\tau(n_2)w$.  So this is in the left hand side.  $\blacksquare$
Now we have natural bijections in both variables
$$\begin{equation*}
\begin{split}
\Hom_G(W,\Ind_P^G \Ind_{M_{\ast}(N_{\ast} \cap M)}^M \pi) & = \Hom_M(W_N,\Ind_{M_{\ast}(N_{\ast} \cap M)}^M \pi) \\ & = \Hom_{M_{\ast}}(W_{N_{\ast}},\pi) \\
& = \Hom_G(W,\Ind_{P_{\ast}}^G\pi)
\end{split}
\end{equation*}$$
so the functors $\Ind_{P_{\ast}}$ and $\Ind_{M_{\ast}(N_{\ast} \cap M)}^M$ must be naturally isomorphic.  
We get the same result if we replace induction by normalized induction.  This is on account of the fact that if $\delta_1$ is the modulus character for $M_{\ast}$ on $(N_{\ast} \cap M)$, and $\delta_2$ is the modulus character for $M$ on $N$, then $\delta_1 \delta_2|_{M_{\ast}}$ is the modulus character for $M_{\ast}$ on $N_{\ast}$.  
