Schemes of a finite projective plane and their extensions

There are several schemes (coherent configurations) associated with a finite projective plane  $\mathcal{P}$. In the paper, a new scheme is constructed, which, in a sense, contains all of them. It turns out that this scheme coincides with the $2$-extension of the nonhomogeneous scheme of  $\mathcal{P}$ and is uniquely determined up to similarity by the order $q$ of  $\mathcal{P}$. Moreover, for $q\ge 3$, the rank of the scheme does not depend on $q$ and equals $416$. The results obtained have interesting applications in the theory of multidimensional extensions of schemes and similarities.