their $m$-subgroup-profile. Here the $m$-subgroup-profile (or $m$-profile) is the multiset of (isomorphism types of) $m$-generated subgroups. Glauberman and Grabowski gave a negative answer by constructing pairs of non-isomorphic groups with the same $\Theta(\sqrt{\log n})$-profiles [@GlaubermanGrabowski]. Subsequently, Wilson constructed many examples of exponent $p$ and nilpotency class $2$ groups which agree in various invariants. In particular they have the same $\Theta(\log n)$-profiles [@WilsonThreshold], which is best possible.

The observation that combinatorial aspects of the group isomorphism problem are less developed is surprising since, for the related graph isomorphism problem, historically, it has been the other way around. Indeed, for graph isomorphism testing, combinatorial approaches are well-developed and often successful, yet their limits have been firmly established. One of the most important tools in this scope is the Weisfeiler-Leman algorithm. The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) iteratively classifies $k$-tuples of vertices of a graph in terms of how they are related to other vertices in the graph. It provides an effective invariant for graph-non-isomorphism (see e.g. [@MR0543783; @DBLP:journals/combinatorica/CaiFI92]). Moreover, $k$-WL can be implemented to run in time $\mathcal{O}(n^{k+1}\log n)$ where $n$ is the number of vertices (see [@MR1060782; @1907.09582]). For fixed $k$, the algorithm is only a partial isomorphism test, in that it 