Given a closed immersion $i: Z \hookrightarrow X$ a coherent sheaf $\mathcal{F}$ on $X$ and a coherent sheaf $\mathcal{G}$ on $Z$, do we have $\mathrm{Ext}^n(\mathcal{F}, i_*\mathcal{G}) = \mathrm{Ext}^n(i^*\mathcal{F}, \mathcal{G})$? For $n = 0$ it is the usual adjunction, so can we deduce it by the usual "universal $\delta$-functor" argument?
Consider the exact $\delta$-functors $\mathrm{Coh}(Z) \to (Ab), \mathcal{G} \mapsto \mathrm{Ext}^n(\mathcal{F}, i_*\mathcal{G})$ and $\mathcal{G} \mapsto \mathrm{Ext}^n(i^*\mathcal{F}, \mathcal{G})$ (exact since $i_*$ is exact). They coincide for $n = 0$, so we just have to check if they are both effacable to coincide for every $n$.
Could we also derive this using derived categories?
Edit: It seems to be wrong: Take $i$ the inclusion of a closed point, then the RHS is always trivial, but I don't think the LHS is. E.g. $\mathrm{Ext}^1(k(x),k(x)) = T_x$ the Zariski tangent space. So where does the above "argument" go wrong?