Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length 2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschildcohomology groups of A and A D(A).