Enumerative geometry of skew lines in ${\mathbb{P}}^3$ with a given associated finite group

For any finite set $\mathcal{L}$ of 3 or more skew lines in ${\mathbb{P}}_{\bar{K}}^3$ over an algebraically closed field $\bar{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal{L}}$ of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\bar{K})$. Given a finite subgroup $G\subset {\mathrm{P}\mathrm{G}\mathrm{L}}_2(\bar{K})$ we study which configurations of lines have $G_{\mathcal{L}}=G$. We derive an upper bound on the number $\mathrm{\mid }\mathcal{L}\mathrm{\mid }$ of lines in terms of the order $\mathrm{\mid }G\mathrm{\mid }$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal{L}$ in ${\mathbb{P}}_{\mathbb{C}}^3$ have $G_{\mathcal{L}}=G$ for certain finite nonabelian groups $G$.