  Let $F$ be a local field with residue characteristic $p$, let $C$ be an
algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a
connected reductive $F$-group. In a previous paper, Florian Herzig and the
authors classified irreducible admissible $C$-representations of
$G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups
of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an
irreducible admissible $C$-representation $\tau$ of $M$, we determine the
lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$ and we show that
$\mathrm{Ind}_P^G \chi \tau$ is irreducible for a general unramified character
$\chi$ of $M$. In the reverse direction, we compute the image by the two
adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation
$\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G
$ respects admissibility, hence coincides with Emerton's ordinary part functor
$\mathrm{Ord}_{\overline{P}}^G$ on admissible representations.
