Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the the semigroup of cofinite monotone partial bijections of the set of pusitive integers $\mathbb{N}$ and $\mathbf{I}\mathbb{N}_{\infty}$ be the semigroup of all partial co-finite isometries of $\mathbb{N}$. We give a criterium when an inverse subsemigroup $S$ of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}$ has the following property: every non-identity congruence of $S$ is a group congrunce. We describe minimal inviverse subsemigroups with such property.

We describe the group $\mathbf{Aut}\left(\mathcal{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathcal{C}_{\mathbb{Z}}$ and study variants $\mathcal{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathcal{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. Especially we prove that $\mathbf{Aut}\left(\mathcal{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers and describe the subset of idempotents $E(\mathcal{C}_{\mathbb{Z}}^{m,n})$ and Green's relations of the semigroup $\mathcal{C}_{\mathbb{Z}}^{m,n}$. Also we show that $E(\mathcal{C}_{\mathbb{Z}}^{m,n})$ is an $\omega$-chain and any two variants of the extended bicyclic semigroup $\mathcal{C}_{\mathbb{Z}}$ are isomorphic.

We describe the group $\mathbf{Aut}\left(\mathcal{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathcal{C}_{\mathbb{Z}}$ and study variants $\mathcal{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathcal{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. Especially we prove that $\mathbf{Aut}\left(\mathcal{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers and describe the subset of idempotents $E(\mathcal{C}_{\mathbb{Z}}^{m,n})$ and Green's relations of the semigroup $\mathcal{C}_{\mathbb{Z}}^{m,n}$. Also we show that $E(\mathcal{C}_{\mathbb{Z}}^{m,n})$ is an $\omega$-chain and any two variants of the extended bicyclic semigroup $\mathcal{C}_{\mathbb{Z}}$ are isomorphic.

We shall discuss on the topic.