  Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we
consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$
is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then
we endow $\bigoplus_{n \ge 0} \textsf{scf}(\mathcal{S}_n)$, the direct-product
of supercharacter function spaces, with the Hopf algebra structure that is
isomorphic to the Hopf algebra $\textsf{QSym}$ of quasisymmetric functions.
Furthermore, we compute the structure constants of the Hopf algebra thus
obtained for the basis consisting of superclass identifier functions. Using our
categorifications, we study a new basis for the Hopf algebra
$\textsf{NSym}_{\mathbb{C}(q,t)}$ of noncommutative symmetric functions over
the rational function field $\mathbb{C}(q,t)$ in commuting variables $q$ and
$t$, with an emphasis on the structure constants of
$\textsf{NSym}_{\mathbb{C}(q,t)}$ for this basis. Some interesting applications
are also obtained via the specializations of $q$ and $t$.
