Certain pairs of irreducible characters of the groups $S_n$ and $A_n$

Investigation of pairs of semiproportional irreducible characters of finite groups is continued. The interest in these investigations is maintained by the discovered earlier connection between the presence or absence of such a pair in a group and the local structure of this group. In the paper, the question of the presence of such pairs in the alternating groups $A_n$ is investigated. A more general problem of description of pairs of irreducible characters of the symmetric group $S_n$ having the same set of roots in one of the sets $A_n$ and $S_n\setminus A_n$ is also considered. All such pairs of irreducible characters of the symmetric group $S_n$ have been found in the case when the main diagonal lengths of the Young diagrams corresponding to these characters do not exceed 2.