the general problem is by Rosenbaum [@DBLP:journals/corr/abs-1304-3935]. Polynomial time algorithms have been developed for various classes of groups [@DBLP:conf/soda/BabaiCGQ11; @DBLP:conf/icalp/BabaiCQ12; @DBLP:conf/stacs/BabaiQ12; @DBLP:conf/csr/DasS19; @DBLP:conf/stacs/Gall09; @DBLP:journals/siamcomp/GrochowQ17; @DBLP:journals/siamcomp/IvanyosQ19; @DBLP:journals/jcst/QiaoNT12]. There is an algorithm running in polynomial time for most orders [@1806.08872]. For the currently fastest isomorphism algorithm for permutation groups see [@DBLP:conf/soda/Wiebking20].

Recent efforts incorporate the Weisfeiler-Leman algorithm into the group isomorphism context [@DBLP:journals/corr/abs-1905-02518; @DBLP:conf/focs/LiQ17]. However, there is a crucial difference to our work. Indeed, in these papers the authors use a combinatorial construction within the groups acting on vector spaces on which the (graph) WL-algorithm is executed. This is different to the general algorithm for all groups defined here. Thus, a priori the two algorithmic approaches are unrelated, warranting further study.

Preliminaries and notation
==========================

#### Groups.

Groups will be denoted by capital Latin characters. For a group $G$ and elements $g,h\in G$ we write their *commutator* as $[g,h]:=ghg^{-1}h^{-1}$ and we use $G'$ to refer to the subgroup of $G$ generated by all commutators. Then $G'$ is the unique minimal normal subgroup of $G$ with abelian quotient. The *centralizer* of $x\in G$ is $C_G(x):=\{g\in G\mid [x,g]=1\}$ and then $Z(G):=\{g\in G\mid C_G(g)=G\}$ is the *center* of $G$. For a prime $p$, a group 