  In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis,
there exists an outer automorphism of the Calkin algebra. (The Calkin algebra
is the algebra of bounded operators on a separable complex Hilbert space,
modulo the compact operators.) In this paper we establish that the analogous
conclusion holds for a broad family of quotient algebras. Specifically, we will
show that assuming the Continuum Hypothesis, if $A$ is a separable algebra
which is either simple or stable, then the corona of $A$ has nontrivial
automorphisms. We also discuss a connection with cohomology theory, namely,
that our proof can be viewed as a computation of the cardinality of a
particular derived inverse limit.
