I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if
$$\iota\_\*:\mathrm{Ext}^\*\_X(\mathcal{F},\mathcal{G}) \to \mathrm{Ext}^\*\_Y (\iota\_\*\mathcal{F},\iota\_\*\mathcal{G})$$ was an iso, respectively if there are certain properties of that map that I may exploit to control said map. 



is the induced map of an embedding an Iso on Ext-groups?
First, there is an adjunction isomorphism
$$
Ext^\bullet(i\_\*F,i\_\*G) \cong Ext^\bullet(Li^\*(i\_\*F),G),
$$
where $Li^\*$ is the derived pullback functor. Furthermore, if $X$ in $Y$ is a locally complete intersection, then
$$
L\_pi^\*(i\_\*F) \cong F \otimes \Lambda^pN^\vee\_{X/Y}.
$$
These two observations combine into a spectral sequence
$$
E\_2^{p,q} = Ext^q(F \otimes \Lambda^pN^\vee\_{X/Y}, G),
$$
whose first row is formed by $Ext^\bullet(F,G)$ and that converges to $Ext^n(i\_\*F,i\_\*G)$.


