An application of Kirchberg's lemma on central sequence algebras to groups of approximately inner automorphisms

We revisit a well-known “surjectivity onto quotient” type lemma of Kirchberg on the central sequence algebra of a separable unital ${\mathrm{C}}^{\ast }$-algebra, and use it to prove a “surjectivity onto quotient” result on approximately inner automorphisms of a separable unital ${\mathrm{C}}^{\ast }$-algebra of stable rank one, which we can partially upgrade also to the non-separable case.