Quadrics on Gushel-Mukai varieties

We study Hilbert schemes of quadrics of dimension $k\in \{0,1,2,3\}$ on smooth Gushel-Mukai varieties $X$ of dimension $n\in \{2,3,4,5,6\}$ by relating them to the relative Hilbert schemes of linear subspaces of dimension $k+1$ of a certain family, naturally associated with $X$, of quadrics of dimension $n-1$ over the blowup of ${\mathbf{P}}^5$ at a point.