We study outer multiplier algebras, C(E)=M(E)/E, also known as corona algebras, and *-homomorphisms A → C(E) . We prove in several instances that for all such maps there must exist an extension to a largerC * -algebra $$A_1 \supseteq A $$ . The Kasparov Technical Theorem gives one class of examples where $$A \cong A_1 \cong C[0,1] \otimes D $$ . Our theorems apply to subhomogeneous C * -algebras, such as $$q\mathbb{C} $$ , the algebra used in Cuntz's picture of K-theory. Where such an extension theorem exists, there must exist an asymptotic morphism $$(\varphi _t ):A_1 \to A $$ whose restriction to A is equivalent to the identity. We also use extension results to prove closure properties for the collection of C *-algebras that have stable relations.