Let G be a normal subgroup of a group E and let Q be a characteristic quotient of G. Then, E maps to Aut(Q) and the image contains Inn(Q). This observation suggests a general strategy for obstructing normal embeddings: find a characteristic quotient Q of G such that every subgroup of Aut(Q) containing Inn(Q) has a property that pulls back through surjective homomorphisms and is incompatible with a potential ambient group E. Conclude that G does not embed as a normal subgroup of E. I will explain how tree actions of automorphism groups and low-generator metabelian quotients implement this strategy, and (time permitting) how factorization theorems for Kähler groups yield further obstructions. Applications include the following: 1. A right-angled Artin group whose finite defining graph and its complement are connected, with more than one vertex, cannot embed as a normal subgroup of Aut(F_n) or Out(F_n) for n \geq 4, or of a Kähler group. 2. If the defining graph is furthermore a tree of even diameter, the obstruction also applies to Aut(F_3), Out(F_3) and mapping class groups of possibly punctured orientable surfaces of genus at least two. 3. A group whose metabelianization is two-generated and not virtually abelian (for example F_2) cannot embed as a normal subgroup of Aut(F_n) or Out(F_n) for n \geq 3, or of those mapping class groups. 4. A finitely generated group G with b_1(G)=1 whose metabelianization is two-generated and not virtually polycyclic cannot embed as a normal subgroup of a Kähler group. This is joint work with Elia Fioravanti and Claudio Llosa-Isenrich.