Classification of semidirect products Given two groups A,B. When is it possible to construct the semidirect product $A \rtimes B$.
Is there a classification for those groups?
How many semidirect products for A,b are there?
 A: Given two groups $A, B$ all one needs to define a semidirect product is a homomorphism $\varphi : B \rightarrow \text{Aut}(A)$. This gives the (external/outer) semidirect product $A \rtimes_{\varphi} B$. 
As a set, $A \rtimes_{\varphi} B = A \times B$. The group operation is given by:
$$ (a_1,b_1)*(a_2,b_2) = (a_1 \varphi(b_1)(a_2) , b_1b_2)$$
Conversely, every semidirect product defines such a homomorphism into the automorphism group...
A: As pointed out in the comment, the post, as it was, described the convention for $A\, {}_φ{\ltimes} B$, not for $A \rtimes_φ B$. So I changed that which explains the difference to @ah11950’s answer.

The semi-direct product can be constructed if you have a group homomorphism $φ \colon A → \operatorname{Aut}(B)$ by setting $A \ltimes B = A × B$ with the group structure given by $(a,b)·(a',b') = (aa',φ(a')(b)b')$.
The motivation behind this is to interpret $φ(a')$ as the conjugation by $a'$, i.e. as $b ↦ a'^{-1}ba'$. And then the equation sort of becomes $ab·a'b' = aa'·(a'^{-1}ba')b' = aa'·φ(a')(b)b'$.
Also I think, this is the convention for semi-direct products writtes as $A \, {}_φ{\ltimes} B$, i.e. the bar and the $φ$ is on the side of the “non-normal subgroup”. Not sure about this, though.
