I am trying to work on a deformation theory problem, but I don't have much experience in it, so any reference or insight would be much appreciated.
Given two coherent sheaves $F$ and $G$ on a projective variety $X$, we can consider an extension $E$ of these two sheaves $0\to F \to E \to G \to 0$. Assume we have two deformations $F'\to F$ and $G' \to G$ (here I am treating deformations as in Hartshorne's book Deformation Theory). I was wondering if there is a way to induce a deformation of $E$ in a "natural" way. For example, if there exists a extension in $Ext^1_{X'}(G',F')$ compatible with the deformation maps, essentially, saying that there exists a surjective map $Ext^1_{X'}(G',F')\to Ext^1_{X}(G,F)\to 0$.
The above result is general in some sense, and I didn't find any reference in this direction. I was trying to apply this in the particular case where $F$ and $G$ are ideal sheaves of points in the $\mathbb{C}$-projective space $\mathbb{P}^3$, and we can even assume that the deformation $G$ is the trivial one, like $G'$ is just the pullback of $G$.
Thanks.