the standard definition of inner semidirect product says that $G$ is a semidirect product of $K$ and $H$ if $G = KH$, for some subgroup $K$ and $H$ of $G$ such that $K$ is normal and $K\cap H =\{H\}$.

Usually wreath product of groups $K\wr H$ is defined as and external semidirect product with $K^n\rtimes_\Phi H$, where $\Phi$ is a homomorphism $H\to Aut(K^n)$.

Is it possible to define an inner wreath product without talking about the homomorphism, i.e., just by decomposing a group into some subgroups satisfying some conditions?

viawhich it acts on $K^n$?) $\endgroup$ – LSpice Feb 26 '19 at 15:42