Linearity criteria for automorphism groups of malabelian groups

Let $G$ be a finitely generated malabelian group, let $A\le \mathrm{O}\mathrm{u}\mathrm{t}(G)$ be a finitely generated subgroup, and let ${\mathrm{\Gamma }}_{G,A}$ denote the preimage of $A$ in $\mathrm{A}\mathrm{u}\mathrm{t}(G)$. We give a general criterion for the linearity of ${\mathrm{\Gamma }}_{G,A}$ in terms of surjections from $G$ to finite simple groups of Lie type.