Linearity criteria for automorphism groups of malabelian groups

Let G be a finitely generated malabelian group, let A≤ \,Out\,(G) A ≤ Out ( G ) be a finitely generated subgroup, and let Γ G,A Γ G , A denote the preimage of A in \,Aut\,(G) Aut ( G ) . We give a general criterion for the linearity of Γ G,A Γ G , A in terms of surjections from G to finite simple groups of Lie type.