Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in [Wikipedia](https://en.wikipedia.org/wiki/Zappa%E2%80%93Sz%C3%A9p_product#External_Zappa%E2%80%93Sz%C3%A9p_products). The actions $\alpha$ and $\beta$ give us a group $A\mathbin{_{\alpha}{\bowtie}_{\beta}} G$ called the [*Zappa-Szép product*][1].

Let $H=A\mathbin{_{\alpha}{\bowtie}_{\beta}} G$ and $H'=A\mathbin{_{\alpha'}{\bowtie}_{\beta'}} G$ be two Zappa-Szép products of $A$ and $G$. Suppose that $\alpha(G)$ and $\alpha'(G)$ are conjugate in Aut(A) and $\beta(G)$ and $\beta'(G)$ are conjugate in Aut(G). Does $H$ and $H'$ are isomorphic.

The answer is yes if $\beta$ and $\beta'$ are trivial actions (See [*math.stackexchange*][2]). But I'm not sure about the generalization of this for Zappa-Szép products. Maybe I need more conditions for the statement to be true. 


  [1]: https://en.wikipedia.org/wiki/Zappa-Sz%C3%A9p_product
  [2]: https://math.stackexchange.com/questions/1093104/isomorphism-of-semidirect-products-df?noredirect=1&lq=1

Any help would be appreciated.