On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$

We study the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ of all endomorphisms of the bicyclic extension $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$. The submonoid $\left\langle\varpi\right\rangle^1$ of $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ with the property that every element of the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ has the unique representation as the product of the monoid endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^2}$ and the element of $\left\langle\varpi\right\rangle^1$ is constructed.