Linearity criteria for automorphism groups of malabelian groups

Let $G$ be a finitely generated malabelian group, let $A\leq\mathrm{Out}(G)$ be a finitely generated subgroup, and let $\Gamma_{G,A}$ denote the preimage of $A$ in $\mathrm{Aut}(G)$.

We give a general criterion for the linearity of $\Gamma_{G,A}$ in terms of surjections from $G$ to finite simple groups of Lie type.