isomorphisms between $\Gamma_1$ and $\Gamma_2$ is $\operatorname{Iso}(\Gamma_1,\Gamma_2)$ and define $\operatorname{Aut}(\Gamma_1):=\operatorname{Iso}(\Gamma_1,\Gamma_1)$.

The *commuting graph* of a group is the graph whose vertices are the group elements and two distinct elements $g,g'$ are adjacent if $[g,g']=1$.

The WL-algorithm for graphs
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Before we explore how the WL-algorithm can be applied to groups, we briefly recapitulate its classic definition for graphs. Given a graph $\Gamma$, the $k$-dimensional version of the algorithm for positive $k\in \mathbb{N}$ repeatedly colors the $k$-tuples of vertices with abstract colors that encode how each tuple is situated within the graph. The initial coloring of each tuple $(g_1,\ldots,g_k)$ encodes the isomorphism type of the graph induced by $\{g_1,\ldots,g_k\}$, taking into account where the vertices occur in the tuple. Specifically, a coloring $\chi_0\colon V(\Gamma)^k\rightarrow S$ into some set of colors is defined so that $\chi_0(g_1,\ldots,g_k) = \chi_0(g'_1,\ldots,g'_k)$ holds exactly if there is an isomorphism from $\Gamma[\{g_1,\ldots,g_k\}]$ to $\Gamma[\{g'_1,\ldots,g'_k\}]$ which sends $g_i$ to $g'_i$ for all $i\in \{1,\ldots,k\}$.

The coloring is now iteratively refined as follows. For a tuple $\bar{g}=(g_1,\ldots,g_k)\in V(\Gamma)^k$ and $x\in V(\Gamma)$, define $\bar{g}_{|i\leftarrow x}$ to be the tuple $(g_1,\ldots,g_{i-1},x,g_{i+1},\ldots,g_k)$ obtained by replacing the $i$-th entry with $x$. Then we define for $k>1$ the coloring $\chi_i(\bar{g}):=$ $$\left( \chi_{i-1}(\bar{g}),{\{\!\!\{}
\newcommand{\rmulset}{\}\!\!\}}(\chi_{i-1}(\bar{g}_{|1\leftarrow x}),\dots,\chi_{i-1}(\bar{g}_{|k\leftarrow 