Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is isomorphism on global sections and let $\mathcal{K}$ be its kernel. Assume that we are given two maps $f, g \in \operatorname{Hom}_{\mathcal{O}}(\mathcal{F}, \mathcal{F}(1))$ and assume that they commute in the sense that $f(1) \circ g - g(1) \circ f = 0$ as a morphism between $\mathcal{F}$ and $\mathcal{F}(2)$. We would like to know, whether the following question has been studied: Does there exist commuting (in the same sense) maps $F,G \in \operatorname{Hom}_{\mathcal{O}}(\mathcal{O}^n,\mathcal{O}(1)^n)$, such that $\pi \circ F = f \circ \pi$ and $\pi \circ G = g \circ \pi$. We may assume that $h^0(\mathcal{K}) = h^1(\mathcal{K}) = h^1(\mathcal{K}(1))$. This seems like something that is known.
Thanks ahead