Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\mathcal{A}\to \mathcal{A}/\mathcal{B}$ is exact. As such, given $X$ and $Y$ two objects in $\mathcal{A}$, there is a group morphism $$ q:\operatorname{Ext}^1_{\mathcal{A}}(X,Y)\to \operatorname{Ext}^1_{\mathcal{A}/\mathcal{B}}(Q(X),Q(Y)) $$ induced by $Q$ on the extension groups. Do we know something on the image or the kernel of $q$?
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