  A binary extended 1-perfect code $\mathcal C$ folds over its kernel via the
Steiner quadruple systems associated with its codewords. The resulting folding,
proposed as a graph invariant for $\mathcal C$, distinguishes among the 361
nonlinear codes $\mathcal C$ of kernel dimension $\kappa$ obtained via
Solov'eva-Phelps doubling construction, where $9\geq\kappa\geq 5$. Each of the
361 resulting graphs has most of its nonloop edges expressible in terms of
lexicographically ordered quarters of products of classes from extended
1-perfect partitions of length 8 (as classified by Phelps) and loops mostly
expressible in terms of the lines of the Fano plane.
