  Using abelian coverings of Salvetti complexes, embeddings of outer
automorphism groups of right-angled Artin groups (RAAGs) into outer
automorphism groups of their particular characteristic subgroups are
constructed. Virtual embeddings of outer automorphism groups of finitely
generated groups having the unique root property into outer automorphism groups
of their particular subgroups are also given. These results provide us with
rich examples of (virtual) embeddings between outer automorphism groups of
RAAGs. Finite subgroups of pure (outer) automorphism groups of RAAGs are also
investigated. $p$-adic valuations and $\mathbb{Z}_p$-ranks of these groups are
determined to establish some non-embeddability conditions.
