Let $Gr(k,n)$ be the Grassmannian of $k$-dimensional vector subspaces $H^k$ of an $n$-dimensional vector space $V$.

Let us fix an $h$-dimensional vector subspace $\Gamma\subset V$ with $h\leq k$, and let $X\subseteq Gr(k,n)$ be the variety parametrizing the $H^k$'s containing $\Gamma$. Then $X\cong Gr(n-k,n-h)$.

Let $E$ the the rank $k-h$ vector bundle on $X$ whose fiber over a point $H^k\in X$ is $H^k/\Gamma$. Does there exists a rank $k-h$ vector bundle $F$ on $Gr(k,n)$ whose restriction to $X$ is $E$ ?