Exact sequences and (semi) direct product Let $0 \longrightarrow H \longrightarrow G  \longrightarrow K \longrightarrow 0$ be a short exact sequence. My question is quite general: how can one recover $G$ knowing $H$ and $K$?


*

*Naive question: is $G$ isomorphic to $ H \times K$? Clearly not, take any non trivial semidirect product.

*Is $G$ isomorphic to a semidirect product between $H$ and $K$? It seems very unlikely since in that case $K$ would have to lift to a subgroup of $G$, but I can't think of a counterexample. 

*Under this last assumption (that $K$ lifts to a subgroup of $H$ on which the restriction of the last arrow induces an isomorphism), is $G$ isomorphic to a semidirect product between $H$ and $K$?  

*More generally, are there classification of short exact sequences under nontrivial assumptions? 
Thanks for your answers!  

Point 4 in full generality is the extension problem, which is a seriously difficult one.
For point $2$, take $G$ to be the cyclic group of order $4$, and $H$ its subgroup of order $2$.
For point 3, yes. (Of course you mean $K$ lifts to a subgroup of $G$.)
An important special case of point 4 is the Schur–Zassenhaus theorem, for the case when $H$ and $K$ have coprime orders.
