This paper addresses and extension of Problem 12.82 of the Kourovka notebook, which asks for all ordered pairs (n,m) such that the symmetric groups Sn embeds in Sm as a maximal subgroup. Problem 12.82 was answered in a previous paper by the author and Benjamin Newton. In this paper, we will consider the extension problem where we allow either or both of the groups from the ordered pair to be an alternating group.
Benesh B. 2008. A classification of certain maximal subgroups of alternating groups. In: Kappe L-C, Magidin A, Morse RF, editors. Contemporary Mathematics 470: Computational Group Theory and the Theory of Groups, AMS Special Session on Computational Group Theory, March 3-4, 2007, Davidson, North Carolina. Providence (RI): American Mathematical Society. p. 21-26.