Liftable automorphisms of right-angled Artin groups

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.