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A supercharacter theory for a finite group $G$ is a set of superclasses eachof which is a union of conjugacy classes together with a set of sums ofirreducible characters called supercharacters that together satisfy certaincompatibility conditions.The aim of this paper is to give a description of somefinite simple groups with exactly three or four supercharacter theories.

Towards the Classification of Finite Simple Groups with exactly Three or Four Supercharacter Theories