I am sorry, but I am quite new to Ext groups of sheaves. However, I have 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.