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Let $\mathscr{F}$ be a family of nonempty inductive subsets of ${\omega}$. Itis proved that an injective endomorphism $\varepsilon$ of the semigroup$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is the transformation if and only if$\varepsilon$ has three distinct fixed points, which is equivalent to existencenon-idempotent element $(i,j,[p))\in\boldsymbol{B}_{\omega}^{\mathscr{F}}$ suchthat $(i,j,[p))\varepsilon=(i,j,[p))$.

On automorphisms of the semigroup  $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ in the case when the family  $\mathscr{F}$ consists of nonempty inductive subsets of ${\omega}$