I would like to ask some basic question about parabolic induction.

Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ where $n=n_1 +n_2$.

Write $G_1=GL_{n_1}(F)$ and let $P_1=M_1N_1$ its parabolic subgroup whose Levis subgroup $M=GL_{a}(F) \times GL_{b}(F)$ where $n_1=a+b$. Similarly, write $G_2=GL_{n_2}(F)$ and let $P_2=M_2N_2$ its parabolic subgroup whose Levis subgroup $M=GL_{c}(F) \times GL_{d}(F)$ where $n_2=c+d$.

Let $\sigma_a,\sigma_b,\sigma_c,\sigma_d$ be irreducible representation of $GL_{a},GL_{b},GL_{c},GL_{d}$ respectively.

Then I am wondering the relation between $\text{Ind}^G_P(\text{Ind}^{G_1}_{P_1} (\sigma_a \boxtimes \sigma_b) \boxtimes \text{Ind}^{G_2}_{P_2} (\sigma_c \boxtimes \sigma_d))$ and $Ind^G_{P_{a,b,c,d}}(\sigma_a \boxtimes \sigma_b \boxtimes \sigma_c \boxtimes \sigma_d )$ where $P_{a,b,c,d}$ is the parabolic subgroup of $G$ whose Levi part is $GL_a \times GL_b \times GL_c \times GL_d$.

I guess there willbe some relation between these under some ideal condition, for example irreducibility of induced representation etc.

Any comments on this will be highly appreciated.