  We develop theory and examples of monoidal functors on tensor categories in
positive characteristic that generalise the Frobenius functor from \cite{Os,
EOf, Tann}. The latter has proved to be a powerful tool in the ongoing
classification of tensor categories of moderate growth, and we demonstrate the
similar potential of the generalisations. More explicitly, we describe a new
construction of the generalised Verlinde categories $Ver_{p^n}$ in terms of
representation categories of elementary abelian $p$-groups. This leads to
families of functors relating to $Ver_{p^n}$ that we conjecture, and partially
show, to exhibit the characteristic properties of the Frobenius functor
relating to $Ver_p$.
