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We study group congruences on the semigroup$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in thecase when an ${\omega}$-closed family $\mathscr{F}$ which consists of inductivenon-empty subsets of $\omega$. It is proven that a congruence $\mathfrak{C}$ on$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is a group congruence if and only ifits restriction on a subsemigroup of $\boldsymbol{B}_{\omega}^{\mathscr{F}}$,which is isomorphic to the bicyclic semigroup, is not the identity relation.Also, all non-trivial homomorphic retracts and isomorphisms of the semigroup$\boldsymbol{B}_{\omega}^{\mathscr{F}}$ are described.

On group congruences on the semigroup  $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the  case when a family $\mathscr{F}$ consists of inductive non-empty subsets of  $\omega$