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Normal Supercharacter Theory

There are two main constructions of supercharacter theories for a group $ G $. The first, defined by Diaconis and Isaacs, comes from the action of a group $A$ via automorphisms on our given group $G$. The second, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup $N$ of $G$ with a supercharacter theory of $G/N$. In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of $G$. We show that when consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of $G$ given by certain values on the central idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.