  We introduce a family of mathematical objects called $\mathcal{P}$-schemes,
where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A
$\mathcal{P}$-scheme is a collection of partitions of the right coset spaces
$H\backslash G$, indexed by $H\in\mathcal{P}$, that satisfies a list of axioms.
These objects generalize the classical notion of association schemes as well as
the notion of $m$-schemes (Ivanyos et al. 2009).
  Based on $\mathcal{P}$-schemes, we develop a unifying framework for the
problem of deterministic factoring of univariate polynomials over finite fields
under the generalized Riemann hypothesis (GRH).
