In answering Classify all groups of order 182 I realized that I don't actually know any good general results that will allow me to determine whether or not two given semidirect products are isomorphic without resorting to various ad-hoc methods (like in that question where I counted elements of order $2$, which by a lucky coincidence was easy and sufficient).
To make the question more specific, let $G$ be a group and $H_1$ and $H_2$ be subgroups of $\rm{Aut}(G)$. What are necessary and/or sufficient conditions for the semidirect products $G \rtimes H_1$ and $G\rtimes H_2$ to be isomorphic?
Clearly if all the groups are finite, we must have $|H_1| = |H_2|$ for them to be isomorphic, but it is not clear to me that we need $H_1\simeq H_2$ unless we further assume that their orders are coprime with $|G|$.
Results that can be applied in a more general setup than the above would also be interesting, as well as results that require a few extra assumptions (like one or more of the groups being abelian or similar).
