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.Comment: 12 page
