On outer automorphisms of certain graph $C^{\ast }$-algebras

Given a countable abelian group $A$, we construct a row finite directed graph $\mathrm{\Gamma }(A)$ such that the $K_0$-group of the graph ${\text{C}}^{\ast }$-algebra ${\text{C}}^{\ast }(\mathrm{\Gamma }(A))$ is canonically isomorphic to $A$. Moreover, each element of $\text{Aut}(A)$ is a lift of an automorphism of the graph ${\text{C}}^{\ast }$-algebra ${\text{C}}^{\ast }(\mathrm{\Gamma }(A))$.