We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free, edge-transitive, twin-free, finite graphs the non-abelian composition factors of the automorphism group have bounded order. We use this to show that the automorphism groups of finite graphs of bounded Hadwiger number are obtained by repeated group extensions using abelian groups, symmetric groups and groups of bounded order.