  A group $G$ is integrable if it is isomorphic to the derived subgroup of a
group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of
$G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if
$G=H'$ for some $H\leq U$. In this work we focus on two problems posed in [1].
We classify the almost-simple finite groups $G$ that are integrable, which we
show to be equivalent to those integrable within $\mathrm{Aut}(S)$, where $S$
is the socle of $G$. We then classify all $2$-homogeneous subgroups of the
finite symmetric group $S_n$ that are integrable within $S_n$.
