  The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful
combinatorial tool in graph isomorphism testing. We address the applicability
of $k$-WL to recognition of graph properties. Let $G$ be an input graph with
$n$ vertices. We show that, if $n$ is prime, then vertex-transitivity of $G$
can be seen in a straightforward way from the output of 2-WL on $G$ and on the
vertex-individualized copies of $G$. However, if $n$ is divisible by 16, then
$k$-WL is unable to distinguish between vertex-transitive and
non-vertex-transitive graphs with $n$ vertices as long as $k=o(\sqrt n)$.
Similar results are obtained for recognition of arc-transitivity.
