In the projective space $\mathrm{PG}(3,q)$, we consider orbits of lines under the stabilizer group of the twisted cubic. In the literature, lines of $\mathrm{PG}(3,q)$ 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.