I work in an Abelian category. If I take the Baer sum $M' + M''$ of two extensions $M'$ and $M''$ of $ M_2$ by $M_1$, i.e., $$ 0 \to M_1 \to M' \to M_2 \to 0$$ is exact, and the same for $M''$, then what do I know about $\mathrm{End}(M' + M'')$? Do I have any access to it? I know everything I want about the other objects (I know $\mathrm{End}(M')$, $\mathrm{End}(M'')$, $\mathrm{End}(M_1)$ and $\mathrm{End}(M_2)$, and also all of their $\mathrm{Hom}$ sets, for instance). I use the definition of Baer sum as successive pullback and pushout of $M' \oplus M''$ by the diagonal and codiagonal maps on $M_2$ and $M_1$, so a more general question is: what do I know about the endomorphism ring of the pullback / pushout of two objects, if I know the objects involved?
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