Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech compactification $\beta\omega$ of $\omega$. The homeomorphism $\bar g$ induces a homeomorphism of the remainder $\omega^*=\beta\omega\setminus\omega$ of the Stone-Cech compactification. So, we obtain a continuous action of the group $G$ on the compact Hausdorff space $\omega^*$. I am interested in properties of the obtained dynamical system $(\omega^*,G)$. Namely, I would like to know the answer to the following

Problem.Is there a subgroup $G\subseteq S_\omega$ such that the dynamical system $(\omega^*,G)$ is topologically transitive (=each nonempty open set has dense orbit) but does not have a dense orbit.

An example of such subgroup $G$ exists under the assumption $\mathrm{non}(\mathcal M)<\mathfrak c$. So, the question actually ask about the situation in ZFC.

**Remark.** If a group $G\subseteq S_\omega$ induces a topologically transitive action on $\omega^*$, then $G$ has large cardinality, namely, $|G|\ge\mathsf \Sigma\ge\max\{\mathfrak b,\mathfrak s,\mathrm{cov}(\mathcal M)\}$. More information on the cardinal $\mathsf \Sigma$ can be found in this preprint.