  Let $\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite
field $\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\mathcal{A}(q)$. On
the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{F}_{q}$,
and let $\mathcal{A} = \mathcal{A}(q) \otimes_{\mathbb{F}_{q}} \Bbbk$. Then $G
= 1+\mathcal{A}$ is an algebraic group over $\Bbbk$ equipped with an
$\mathbb{F}_{q}$-rational structure given by the usual Frobenius map $F:G\to
G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{F}$. For every
$n \in \mathbb{N}$, the $n$th power $F^{n}:G\to G$ is also a Frobenius map, and
$G^{F^{n}}$ identifies with $G(q^{n}) = 1 + \mathcal{A}(q^{n})$. The Frobenius
map restricts to a group automorphism $F:G(q^{n})\to G(q^{n})$, and hence it
acts on the set of irreducible characters of $G(q^{n})$. Shintani descent
provides a method to compare $F$-invariant irreducible characters of $G(q^{n})$
and irreducible characters of $G(q)$. In this paper, we show that it also
provides a uniform way of studying supercharacters of $G(q^{n})$ for $n \in
\mathbb{N}$. These groups form an inductive system with respect to the
inclusion maps $G(q^{m}) \to G(q^{n})$ whenever $m \mid n$, and this fact
allows us to study all supercharacter theories simultaneously, to establish
connections between them, and to relate them to the algebraic group $G$.
Indeed, we show that Shintani descent permits the definition of a certain
``superdual algebra'' which encodes information about the supercharacters of
$G(q^{n})$ for $n \in \mathbb{N}$.
