fully faithfulness of the direct image, but on $\text{Ext}^1$

Let $i:C\subset S$ where $S$ a smooth projective complex surface and $C$ a smooth projective complex curve, then the functor $i_*:\textbf{Coh}(C)\rightarrow\textbf{Coh}(S)$ is fully faithful, i.e. $$\text{Hom}_C(\mathcal{E},\mathcal{F})\cong\text{Hom}_S(i_*\mathcal{E},i_*\mathcal{F})$$ Can we show that $\text{Ext}^1_C(\mathcal{E},\mathcal{F})\cong\text{Ext}^1_S(i_*\mathcal{E},i_*\mathcal{F})$ in general or with some additional (but not vary special) conditions?

This post Significance of adjoint relationship with Ext instead of Hom may be related.

To compare $\mathrm{Ext}^1(E,F)$ to $\mathrm{Ext}^1(i_*E,i_*F)$ one can proceed as follows. First, there is adjunction isomorphism $$ \mathrm{Ext}^\bullet(i_*E,i_*F) \cong \mathrm{Ext}^\bullet(Li^*i_*E,F). $$ Next, since $i$ is an embedding of a Cartier divisor, there is a distinguished triangle $$ Li^*i_*E \to E \to E(-C)[2], $$ where $E(-C) = E \otimes \mathcal{O}_C(-C)$. Applying the functor $\mathrm{Ext}^\bullet(-,F)$ to this triangle, we obtain a long exact sequence $$ \dots \to \mathrm{Ext}^{i-2}(E(-C),F) \to \mathrm{Ext}^{i}(E,F) \to \mathrm{Ext}^{i}(Li^*i_*E,F) \to \mathrm{Ext}^{i-1}(E(-C),F) \to \dots $$ Since $\mathrm{Ext}^i(-,-)$ between two sheaves vanishes for $i \not\in \{0,1,2\}$, using the above adjunction isomorphisms, we obtain an isomorphism $$ \mathrm{Hom}(E,F) \to \mathrm{Hom}(i_*E,i_*F), $$ a short exact sequence $$ 0 \to \mathrm{Ext}^{1}(E,F) \to \mathrm{Ext}^{1}(i_*E,i_*F) \to \mathrm{Hom}(E(-C),F) \to 0 $$ and yet another isomorphism $$ \mathrm{Ext}^{2}(i_*E,i_*F) \to \mathrm{Ext}^1(E(-C),F). $$ In particular, the morphism $\mathrm{Ext}^{1}(E,F) \to \mathrm{Ext}^{1}(i_*E,i_*F)$ is an isomorphism if and only if $\mathrm{Hom}(E(-C),F) = 0$.

EDIT: The argument below is wrong, $\text{Ext}^i(\mathcal{F},-)$ is not computed using flasque resolutions. As a counterexample, a skyscraper sheaf is always flasque, but $\text{Ext}^1(k(x),k(x))$ typically does not vanish.

This is true with no further assumptions.

In this case $i_*$ is an exact functor. You compute $\text{Ext}^1_C(\mathcal{E},\mathcal{F})$ by taking a flasque resolution of $\mathcal{F}$, applying $\text{Hom}(\mathcal{E},-)$ and taking the first cohomology. Now if you apply $i_*$ to this resolution you get a flasque resolution (check that $i_*$ preserves flasque sheaves!). So $\text{Ext}^1_S(i_*\mathcal{E},i_*\mathcal{F})$ may be computed by applying the functor $\text{Hom}(i_*\mathcal{E},i_*(-))$ to your original flasque resolution.

If you know about effaceable functors, you can just argue that $\text{Ext}^k_S(i_*\mathcal{E},i_*(-))$ vanishes on flasque sheaves.