A statement that I find sometimes in different forms and it puzzles me. I call $\mathbb G(1,N)$ the Grassmannian, seen as a submanifold of $\mathbb{P}(\bigwedge^2 \mathbb{C}^{N+1})$, and $H^l$ a codimension-$l$ linear subspace inside $\mathbb{P}(\bigwedge^2 \mathbb{C}^{N+1})$.
An example from https://arxiv.org/pdf/math/9904060.pdf, proof of Theorem $1.2$: "We prove first that all automorphisms of $\mathbb G(1,N) \cap H^l$ are induced by automorphisms of $\mathbb G(1,N)$. This follows immediately, once we show that all divisors of $\mathbb G \cap H^l$ are induced by divisors of $\mathbb{P}(\bigwedge^2 \mathbb{C}^{N+1})$."
In the case of a projective space instead of the Grassmannian $\mathbb G(1,N)$ I would have understood the argument, since I would have said that the embedding associated to a very-ample divisor would have been equivariant with respect to the linear action of the automorphism on the global sections and the automorphism itself. But I don't know any generalization that involves Grassmannians. Do you have any references? Necessary and sufficient conditions for this to happen? Thank you!