  A graph is $\textit{cubelike}$ if it is a Cayley graph for some elementary
abelian $2$-group $\mathbb{Z}_2^n$. The core of a graph is its smallest
subgraph to which it admits a homomorphism. More than ten years ago,
Ne\v{s}et\v{r}il and \v{S}\'amal (On tension-continuous mappings.
$\textit{European J. Combin.,}$ 29(4):1025--1054, 2008) asked whether the core
of a cubelike graph is cubelike, but since then very little progress has been
made towards resolving the question. Here we investigate the structure of the
core of a cubelike graph, deducing a variety of structural, spectral and
group-theoretical properties that the core "inherits" from the host cubelike
graph. These properties constrain the structure of the core quite severely ---
even if the core of a cubelike graph is not actually cubelike, it must bear a
very close resemblance to a cubelike graph. Moreover we prove the much stronger
result that not only are these properties inherited by the core of a cubelike
graph, but also by the orbital graphs of the core. Even though the core and its
orbital graphs look very much like cubelike graphs, we are unable to show that
this is sufficient to characterise cubelike graphs. However, our results are
strong enough to eliminate all non-cubelike vertex-transitive graphs on up to
$32$ vertices as potential cores of cubelike graphs (of any size). Thus, if one
exists at all, a cubelike graph with a non-cubelike core has at least $128$
vertices and its core has at least $64$ vertices.
