Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.

Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.

Assume all obstructions classes belonging to $F$ vanish, e.g. $Ext^2(F,F)=0$ and we have a nontrivial element $x \in Ext^1(F,F)$, i.e. a deformation $G$ of $F$ over $k[\epsilon]$.

Can we extend this deformation to a curve? More precisely:

Can we find a deformation $\mathcal{F}$ of $F$ over a smooth connected curve $C$, such that $\mathcal{F}$ gives back $G$ under the pullback induced by $Spec(k[\epsilon])\rightarrow C$?

If yes, is this merely an existence result or can we construct the sheaf $\mathcal{F}$ and the curve $C$?