  A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion
from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We
give a new proof of their result, which enables us to generalize it to the case
where $N=3$. More generally, we give sufficient conditions on a subgroup
$\Gamma$ of $\mathrm{Out}(F_N)$ ensuring that its abstract commensurator
$\mathrm{Comm}(\Gamma)$ is isomorphic to its relative commensurator in
$\mathrm{Out}(F_N)$. In particular, we prove that the abstract commensurator of
the Torelli subgroup $\mathrm{IA}_N$ for all $N\ge 3$, or more generally any
term of the Andreadakis--Johnson filtration if $N\ge 4$, is equal to
$\mathrm{Out}(F_N)$. Likewise, if $\Gamma$ the kernel of the natural map from
$\mathrm{Out}(F_N)$ to the outer automorphism group of a free Burnside group of
rank $N\geq 3$, then the natural map
$\mathrm{Out}(F_N)\to\mathrm{Comm}(\Gamma)$ is an isomorphism.
