Let $\mathcal{C}$ be an abelian category (feel free to put more adjectives here) and $\mathcal{D}$ a full abelian subcategory closed under kernels and cokernels.
Then by definition for $A,B\in \mathcal{D}$, $\mathrm{Hom}_{\mathcal{D}}(A,B)\simeq \mathrm{Hom}_{\mathcal{C}}(A,B)$.
Moreover, since the subcategory is full, an extension $0\to B\to X\to A\to 0$ with $X\in \mathcal{D}$ splits in $\mathcal{D}$ if and only if it splits in $\mathcal{C}$, so we have that $$\mathrm{Ext}^1_{\mathcal{D}}(A,B)\subseteq \mathrm{Ext}^1_{\mathcal{C}}(A,B).$$
Can we somehow argue this way and conclude that $\mathrm{Ext}^i_{\mathcal{D}}(A,B)\subseteq \mathrm{Ext}^i_{\mathcal{C}}(A,B)$ for all $i$?
