On the group of automorphisms of the semigroup ${\mathit{B}}_{\mathbb{Z}}^{\mathcal{F}}$ with the family $\mathcal{F}$ of inductive nonempty subsets of $\omega$

We study automorphisms of the semigroup ${\mathit{B}}_Z^{\mathcal{F}}$ with the family $\mathcal{F}$ of inductive nonempty subsets of $\omega$ and prove that the group $\mathbf{A}\mathbf{u}\mathbf{t}({\mathit{B}}_{\mathbb{Z}}^{\mathcal{F}})$ of automorphisms of the semigroup ${\mathit{B}}_Z^{\mathcal{F}}$ is isomorphic to the additive group integers.