Properties of covers in the lattice of group topologies for nilpotent groups

A nilpotent group $\widehat G$ and two group topologies $\widehat\tau''$ and $\widehat\tau*$ on $\widehat G$ are constructed such that $\widehat\tau*$ is a coatom in the lattice of all group topologies of the group $\widehat G$ and such that between $\inf\{\widehat\tau'',\widehat\tau_d\}$ and $\inf\{\widehat\tau'',\widehat\tau*\}$ there exists an infinite chain of group topologies.