In his 92 paper, Mukai showed that a general genus $6$ curve may be represented in $\mathbb{P}^9$ as the intersection of the Grassmannian $G(2,5)$ (under the Plucker embedding), a plane $H\cong \mathbb{P}^5$ and a quadric hypersurface $Q$ on $H$.

I would like to understand the uniqueness properties of this construction. Let us fix the embedding $G(2,5)\subseteq \mathbb{P}^9$, and lets consider two models, corresponding to $(H,Q)$ and $(H',Q')$. Is it true that $H=H'$ and $Q=Q'$? Or maybe they belong to the same orbit under some group action (in this case, the action should respect $G(2,5)$ though)?

There is an obvious redundancy, i.e., $Q$ can be changed by $Q+$ a linear combination of the quadrics defining $G(2,5)$, so I am considering the space of quadrics modulo those defining $G(2,5)$ (which is a five-dimensional vector space).