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. The actions $\alpha$ and $\beta$ give us a group $A\mathbin{_{\alpha}{\bowtie}_{\beta}} G$ called the Zappa-Szép product.

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). 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.

Any help would be appreciated.

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. The actions $\alpha$ and $\beta$ give us a group $A\mathbin{_{\alpha}{\bowtie}_{\beta}} G$ called the Zappa–Szép product.

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 $\operatorname{Aut}(A)$ and $\beta(G)$ and $\beta'(G)$ are conjugate in $\operatorname{Aut}(G)$. Must $H$ and $H'$ be isomorphic?

The answer is yes if $\beta$ and $\beta'$ are trivial actions (See math.stackexchange). 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.

Any help would be appreciated.