Do permutation isomorphic actions produce isomorphic semidirect products? Let $H$ and $K$ be two groups and suppose


*

*$H$ acts on $K$.

*$H$ preserves the group structure of $K$.

*$\phi$ is the permutation representation of the action:


$\phi:H\rightarrow Aut(K)$
$\phi(x)(k)=k^x$
Then the set $K\times H$ becomes a group with the product
$(a,x)(b,y)=(ab^{x^{-1}},xy)$. This group is denoted by $K\rtimes_\phi H$.
Do two permutation isomorphic (group-structure-preserving) actions from $H$ on $K$, produce isomorphic groups $K\rtimes H$?
How about equivalent actions?

if $K\rtimes_\phi H$ and $K\rtimes_\psi H$ are isomorphic are the actions represented by $\phi$ and $\psi$ isomorphic?

Since you are not requiring your bijection $\lambda:K \to K$ to be a group isomorphism, the answer to your first two questions is no. I can think of a counterexample in which $K$ is cyclic of order 8 and $H$ has order 2.
The answer to the final question is also no. Let $G = K \times H$, with $K$ dihedral of order $2n$ for some $n>2$, and $H$ cyclic of order 2. By choosing a different complement of $K$ in $G$, we can also express $G$ as $K \rtimes H$ with nontrivial action of $H$ on $K$.
