On some questions related to integrable groups

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H^{\mathrm{\prime }}\simeq G$, 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 $G=H^{\mathrm{\prime }}$ for some $H\le U$. 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 $\mathrm{A}\mathrm{u}\mathrm{t}(S)$, 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 $S_n$.