Supercharacter theories of semiextraspecial p-groups and Frobenius groups

In this paper, we describe all supercharacter theories of semiextraspecial p-groups and Frobenius groups. We first review the constructions which will classify these supercharacter theories. We then show that the supercharacter theories of these groups, as well as the more general Camina pairs, must have nontrivial C-normal subgroups. Next, we examine the superclass structure to show that the supercharacter theory can be constructed from supercharacter theories of subgroups of the group and its quotient groups. We can apply this to count the number of supercharacter theories for the group.