  In this work, we present a generalization of extended persistent homology to
filtrations of graded sub-groups by defining relative homology in this setting.
Our work provides a more comprehensive and flexible approach to get an
algebraic invariant overcoming the limitations of the standard approach. The
main contribution of our work is the development of a stability theorem for
extended persistence modules using an extension of the definition of
interleaving and the rectangle measure. This stability theorem is a crucial
property for the application of mathematical tools in data analysis. We apply
the stability theorem to extended persistence modules obtained from extended
path homology of directed graphs and extended homology of hypergraphs, which
are two important examples in topological data analysis.
