Given $G \in SU(4)$, what are the level sets of the function $F(V) = |tr(G^{\dagger}V)|^2$?

Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in some basis?

Cross-posted after no answer on MSE.

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$?

Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in some basis?

Cross-posted after no answer on MSE.