Supercharacter theory of an algebra group

Definition
Suppose $$F$$ is a finite field of size $$q$$, a prime power with underlying prime $$p$$. Suppose $$N$$ is a nilpotent associative finite-dimensional algebra over $$F$$. Suppose, further, that $$G$$ is the algebra group corresponding to $$N$$, i.e., $$G = 1 + N$$..

The supercharacter theory of $$G$$ is a supercharacter theory defined as follows.

Superconjugacy classes
Note that $$G$$ is a subgroup of the unitization $$N + F$$, hence acts on $$N + F$$ by both left and right multiplication. The ideal $$N$$ is invariant under both actions, so we get a left and a right $$G$$-action on $$N$$ that commute with each other. Consider the double orbits of $$N$$ under these actions, i.e., sets of the form $$GxG, x \in N$$. The corresponding sets $$1 + GxG$$, which are subsets of $$G$$, are the superconjugacy classes.

Note that the conjugation action of $$G$$ on $$N$$ corresponds to its conjugation action on itself, and the orbits for this conjugation action are contained inside the double orbits, so the superconjugacy classes are unions of conjugacy classes.