Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

We prove that every right-angled Artin group occurs as a finite-index subgroup of the outer automorphism group of another right-angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^N$ for some $N$. For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day-Wade and Wade-Br\"uck about when this group is a right-angled Artin group and when it has finite index.