I know there are methods of showing when $H\rtimes_{\Psi_{1}}K$$\cong$$H\rtimes_{\Psi_{2}} K$.
However, what about semidirect products in which the H's differ.
Is it ever the case where $|H_{1}|=|H_{2}|$, $H_{1} \ncong H_{2}$ but we have that $H_{1}\rtimes_{\Psi_{1}}K$$\cong$$H_{2}\rtimes_{\Psi_{2}} K$.
An example is the dihedral group of order $8$. Its elements are the rotations $1,a,a^2,a^3$, and the reflections $b,ab,a^2b,a^3b$.
We can take $H_1=\{1,a,a^2,a^3\}$, $H_2=\{1,a^2,ab,a^3b\}$, with $K=\{1,b\}$. Then $H_1$ is cyclic and $H_2$ is a Klein $4$-group.