Suppose $$\xi: 0\to B\to X\to A\to 0$$ is a short exact sequence of R-modules, where $R$ is a commutative ring.
Applying the Ext functors $\textrm{Ext}^*(A,-)$ to $\xi$, we get an exact sequence $$ \textrm{Hom}(A,X)\to \textrm{Hom}(A,A) \stackrel{\partial}{\to} \textrm{Ext}^1(A,B), $$ and $\xi$ is split if and only if $\partial\,(\textrm{id}_A)=0\in \textrm{Ext}^1(A,B).$
On the other hand, applying $\textrm{Ext}^*(-,B)$ to $\xi$, we get an exact sequence $$ \textrm{Hom}(X,B)\to \textrm{Hom}(B,B) \stackrel{\partial'}{\to} \textrm{Ext}^1(A,B), $$ and $\xi$ is split if and only if $\partial'\,(\textrm{id}_B)=0\in \textrm{Ext}^1(A,B).$
Is $\partial\,(\textrm{id}_A)$ always equal to $\partial'\,(\textrm{id}_B)$?