take $A$ and $B$ two subgroups of THE SAME group $G$. In this post i'm not interested in the general case of $A$ and $B$ being given groups not necessarely subgroups of the same group.
if $A\cap B =\{1\}$ and $A$ is normalized by $B$ then the set $AB=\{ab\,|a\in A, b\in B\} $ is a subgroup of $G$ that we call the internal semidirect product of $A$ by $B$.
Now for each action $\phi:B\rightarrow Aut(A)$ we can endow the cartesian product $A\times B$ with the group structure $(a_1,b_1)*(a_2,b_2)=(a_1\phi_{b_1}(a_2),b_1b_2)$ and we get a group called the external semidirect product of $B$ acting on $A$ corresponding to $\Phi$. Why we only care about the action of conjugation $\phi_b(a)=bab^{-1}$? I know only one reason: when acting by conjugation, the external semidirect product $A\times B$ is ISOMORPHIC to the internal semidirect product $AB$. But what about other actions? are there situations where external semidirect products of subgroups $A$ and $B$ of the same group $G$ with action other than conjugation are considered?