  There is a long tradition of categorifying combinatorial Hopf algebras by the
modules of a tower of algebras (or even better via the representation theory of
a tower of groups). From the point of view of combinatorics, such a
categorification supplies canonical bases, inner products, and a natural avenue
to prove positivity results. Recent ideas in supercharacter theory have made
fashioning the representation theory of a tower of groups into a Hopf structure
more tractable. This paper applies such a program to the Malvenuto--Reutenauer
Hopf algebra. In particular, we design functors on the representation theory of
a tower of $p$-groups that realize the Hopf structure of the
Malvenuto--Reutenauer algebra in such a way that its well-known fundamental
basis corresponds to a supercharacter basis.
