I want to ask some basic two questions on the parabolic induction.

Let $F$ be a local fields.

Let $\chi_1,\chi_2$ be two characters of $GL_1(F)$ and $GL_1 \times GL_1$ be the Levi part of the parabolic subgroup $P$ of $GL_2$. Let $v$ be the character of $GL_2$ composed with the determinant map and the absolute value.

Then which representation is isomorphic to $(\text{Ind}_P^{GL_2} (\chi_1 \boxtimes \chi_2)) \cdot v$ or its quotient? Is it $\text{Ind}_P^{GL_2} (\chi_1|\cdot|\boxtimes \chi_2|\cdot|)$?

Consider $\text{Ind}_{P_{(1,2,1)}}^{GL_4} (\chi_1|\cdot| \boxtimes \text{Ind}_{P}^{GL_2}(\chi_1 \boxtimes \chi_2) \boxtimes \chi_1 |\cdot|^{-1})$. Then is its irreducible quotients isomorphic to $\text{Ind}_{P_{(3,1)}}^{GL_4} (\chi_1 \circ \det_{GL_3}\boxtimes \chi_2))$?

Any comments on this will be highly appreciated!