Suppose I have groups $A,B$ and $C$ for which the following information is given:

1) The group $G_{AB}$ is a central extension of $B$ by $A$, where the abelian group $B$ acts trivially

2) The group $G_{BC}$ is an extension of $C$ by $B$, where $C$ does *not* act trivially, and the action is given by $\theta: C \rightarrow Aut(B)$.

Then, under what conditions can we make the following statement?

There exists a group $G_{ABC}$ which is an extension of $G_{BC}$ by $A$ as well as an extension of $C$ by $G_{AB}$, where the respective actions are induced by the actions given in (1) and (2).

In other words, the following diagram needs to be filled in by groups such that every row and column is a group extension (thanks to Neil Strickland for making this precise): $\require{AMScd}$ \begin{CD} A @>>> G_{AB} @>>> B \\ @| @VVV @VVV \\ A @>>> G_{ABC} @>>> G_{BC} \\ & @VVV @VVV \\ && C @= C \end{CD}

Furthermore, when is $G_{ABC}$ unique?

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