Baumann-components of finite groups of characteristic p, reduction theorems

We continue the project started in "Baumann-components of finite groups of characteristic p, general theory" to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Baup(G). The reduction theorem proved in that paper allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G/Op(G) and the action of G on Ω1(Z(Op(G))). In addition, we prove reduction theorems which allow to focus on groups G which satisfy G/Op(G)≅SLn(q), Sp2n(q) or G2(q) and Op(G)⩽B for B∈Baup(G).