  In this paper we combine many of the standard and more recent algebraic
techniques for testing isomorphism of finite groups (GpI) with combinatorial
techniques that have typically been applied to Graph Isomorphism. In
particular, we show how to combine several state-of-the-art GpI algorithms for
specific group classes into an algorithm for general GpI, namely: composition
series isomorphism (Rosenbaum-Wagner, Theoret. Comp. Sci., 2015; Luks, 2015),
recursively-refineable filters (Wilson, J. Group Theory, 2013), and low-genus
GpI (Brooksbank-Maglione-Wilson, J. Algebra, 2017). Recursively-refineable
filters -- a generalization of subgroup series -- form the skeleton of this
framework, and we refine our filter by building a hypergraph encoding low-genus
quotients, to which we then apply a hypergraph variant of the k-dimensional
Weisfeiler-Leman technique. Our technique is flexible enough to readily
incorporate additional hypergraph invariants or additional characteristic
subgroups.
