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′≃ 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≤ U. In this work we focus on two problems posed in Araújo et al. (Israel J Math 234:149–178, 2019). We classify the almost-simple finite groups G that are integrable, which we show to be equivalent to those integrable within Aut(S), where S is the socle of G. We then classify all 2-homogeneous subgroups of the finite symmetric group Sn that are integrable within Sn.