On the semigroup of endomorphisms of the semigroup ${\mathit{B}}_{\omega }^{{\mathcal{F}}^2}$ with the two-element family ${\mathcal{F}}^2$ of inductive nonempty subsets of $\omega$

We study the semigroup $\stackrel{‾}{}({\mathit{B}}_{\omega }^{{\mathcal{F}}^2})$ of all endomorphisms of the bicyclic extension ${\mathit{B}}_{\omega }^{{\mathcal{F}}^2}$ with the two-element family ${\mathcal{F}}^2$ of inductive nonempty subsets of $\omega$. The submonoid ${⟨\varpi ⟩}^1$ of $\stackrel{‾}{}({\mathit{B}}_{\omega }^{{\mathcal{F}}^2})$ with the property that every element of the semigroup $\stackrel{‾}{}({\mathit{B}}_{\omega }^{{\mathcal{F}}^2})$ has the unique representation as the product of the monoid endomorphism of ${\mathit{B}}_{\omega }^{{\mathcal{F}}^2}$ and the element of ${⟨\varpi ⟩}^1$ is constructed.