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We study automorphisms of the semigroup$\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ with the family $\mathscr{F}$ ofinductive nonempty subsets of $\omega$ and prove that the group$\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms ofthe semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ is isomorphic to theadditive group integers.

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