Does there exist a pair of finite groups $G$ and $H$ satisfying both of the short exact sequences $1 \rightarrow G \rightarrow H \rightarrow A\_4 \rightarrow 1$ and $1 \rightarrow G \rightarrow H \rightarrow D\_6 \rightarrow 1$? Of course the homomorphisms $G \to H$ in these short exact sequences are not the same.



Pair of short exact sequences of groups
Call two finite groups $Q\_1$ and $Q\_2$ *compatible* if there exists a finite group $G$ with two isomorphic normal subgroups $N\_1$ and $N\_2$ such that $G/N\_i\cong Q\_i$.


One can show the following:


**Proposition:**
If two groups are compatible, then they have subnormal series of the same length with the same factor groups appearing in the same order.


*Proof:*
Let $Q\_1$ and $Q\_2$ be compatible with $G$ a witness of minimal order, and $N\_1$ and $N\_2$ the two corresponding isomorphic normal subgroups and let $\alpha$ be an isomorphism from $N\_1$ to $N\_2$. Let $M=N\_1\cap N\_2$. Note that $M$ and $\alpha(M)$ are isomorphic and normal in $N\_2$, so $N\_2/M$ and $N\_2/\alpha(M)$ are compatible, with $N\_2$ as a witness.


But $N\_2/M\cong N\_1N\_2/N\_1$ while $N\_2/\alpha(M)\cong N\_1/M\cong N\_1N\_2/N\_2$. Minimality of $G$ implies that $N\_1N\_2<G$, so that $Q\_1$ and $Q\_2$ have $G/N\_1N\_2$ as a non-trivial common quotient, but moreover the corresponding normal subgroups are compatible, so the result follows by induction.
$\square$


I've read somewhere that the above argument (which is in some sense a generalisation of the one by Robert) is due to Sims, but I'm not sure the argument itself is actually written anywhere.


In particular, it shows that $A\_4$ and $D\_6$ are not compatible, because they don't have such subnormal series. (Any series for $A\_4$ has a $C\_3$ "on top", and in $D\_6$, a $C\_2$ "on top".)


I've been interested in the question of determining which groups are compatible for a while. I think it's an interesting question and the answer is not known. See


Giudici, Glasby, Li, Verret, Arc-transitive digraphs with quasiprimitive local actions, Journal of Pure and Applied Algebra 223 (2019) 1217-1226


for some motivation and further results.


See also <https://math.stackexchange.com/questions/4295186/which-pairs-of-groups-are-quotients-of-some-group-by-isomorphic-subgroups/4296206#4296206>


