  Let $N$ be at least 4. We prove that every injective homomorphism from the
Torelli subgroup into $Out(F_N)$ differs from the inclusion by a conjugation in
$Out(F_N)$. This applies more generally to the following subgroups: every
finite-index subgroup of $Out(F_N)$ (recovering a theorem of Farb and Handel);
every subgroup that contains a finite-index subgroup of one of the groups in
the Andreadakis--Johnson filtration; every subgroup that contains a power of
every linearly-growing automorphism; more generally, every twist-rich subgroup
(subgroups that contain sufficiently many twists in an appropriate sense).
  Among applications, this recovers the fact that the abstract commensurator of
every group above is equal to its relative commensurator in $Out(F_N)$; it also
implies that all subgroups in the Andreadakis--Johnson filtration are
co-Hopfian.
  We also prove the same rigidity statement for subgroups of $Out(F_3)$ which
contain a power of every Nielsen transformation. This shows in particular that
$Out(F_3)$ and all its finite-index subgroups are co-Hopfian, extending a
theorem of Farb and Handel to the $N=3$ case.
