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.