It's a long time I am not working with groups. Maybe you know the answer to these questions.
Suppose that $1\rightarrow A\xrightarrow{i_1} B_1\xrightarrow{\pi_1}C\rightarrow 1$ and $1\rightarrow A\xrightarrow{i_2} B_2\xrightarrow{\pi_2} C\rightarrow 1$ are two short exact sequences of (not necessarily abelian) groups. Suppose we know that they are isomorphic.
a) Is there an isomorphism $\beta:B_1\rightarrow B_2$ such that $(id_A,\beta,id_C)$ is an isomorphism between the two exact sequences? ( I mean, such that $\beta i_1=i_2$ and $\pi_1=\pi_2\beta$).
b) If no, then what about the case where the sequences split?
Thanks in advance.
If I am understanding your question correctly, then the answer is no even for split extensions. For example, $A$ could be the additive group of a vector space on which the action of $C$ makes $A$ into a $C$-module. There could be an outer automorphism of $C$ that does not preserve the module. In that situation, if your given isomorphism between the exact sequences involved such an outer automorphism of $C$, then there would be no isomorphism $B_1 \to B_2$ that induced the identity on $C$.
As a specific example, take $C={\rm PSL}_3(2)$ with $N$ elementary abelian of order $8$, and $C$ inducing the natural module action, and use the duality outer automorphism of $C$. This does not preserve the module but maps it to its dual.