  Given a regular covering map $\varphi:\Lambda \to \Gamma$ of graphs, we
investigate the subgroup $\operatorname{LAut}(\varphi)$ of the automorphism
group $\operatorname{Aut}(A_\Gamma)$ of the right-angled Artin group
$A_\Gamma$. This subgroup comprises all automorphisms that can be lifted to
automorphisms of $A_\Lambda$. We first show that $\operatorname{LAut}(\varphi)$
is generated by a finite subset of Laurence's elementary automorphisms.
  For the subgroup $\operatorname{FAut}(\varphi)$ of
$\operatorname{Aut}(A_\Lambda)$, which consists of lifts of automorphisms in
$\operatorname{LAut}(\varphi)$, there exists a natural homomorphism
$\operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi)$ induced by
$\varphi$. We then show that the kernel of this homomorphism is virtually a
subgroup of the Torelli subgroup $\operatorname{IA}(A_\Lambda)$ and deduce a
short exact sequence reminiscent of results from the Birman--Hilden theory for
surfaces.
