Orbits of the class ${\mathcal{O}}_6$ of lines external with respect to the twisted cubic in $\mathrm{P}\mathrm{G}(3,q)$

In the projective space $\mathrm{P}\mathrm{G}(3,q)$, we consider orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of $\mathrm{P}\mathrm{G}(3,q)$ are partitioned into classes, each of which is a union of line orbits. We propose an approach to obtain orbits of the class named ${\mathcal{O}}_6$, whose complete classification is an open problem. For all even and odd $q$ we describe a family of orbits of ${\mathcal{O}}_6$ and their stabilizer groups. The orbits of this family include an essential part of all ${\mathcal{O}}_6$ orbits.