Let $A$ and $G$ be two groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ be a group homomorphism and $\beta: A\rightarrow\operatorname{Bij}(G)$ an anti-homomorphism 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 $\beta$ induce a group $A\mathbin{\_{\alpha}{\bowtie}\_{\beta}} G$ called the [*Zappa-Szép product*](https://en.wikipedia.org/wiki/Zappa-Sz%C3%A9p_product). This group is a natural generalization of the semidirect product of two groups in which neither of the factors is required to be normal. I know some examples of semidirect products induced by nontrivial actions and isomorphic to direct products. However, I couldn't find a group $H=A\mathbin{\_{\alpha}{\bowtie}\_{\beta}} G$ isomorphic to the direct product $A\times G$ where the actions $\alpha$ and $\beta$ are both nontrivial. So I will be thankful if someone provides me an example. Notice that I have asked a similar question in [*math.stackexchange*](https://math.stackexchange.com/questions/4751563/zappa-sz%c3%a9p-product-vs-direct-product).



Is it possible for a direct product to be isomorphic to the Zappa–Szép product?
Let $S$ be a nonabelian group with a fixed-point-free automorphism $\alpha$. (Such groups for which $\alpha$ has prime order are necessarily nilpotent. I think the smallest example is the nonabelian group of order $7^3$ and exponent $7$, which has a fixed-point-free automorphism of order $3$.)


Now let $G = S \times S$ and define the (diagonal) subgroups $H$ and $K$ of $G$ by
$$ H = \{ (s,s) : s \in S \},\quad K = \{(s,\alpha(s)) : s \in S \}.$$
Then $H \cong K \cong S$, $H \cap K = \{1\}$ and $G = HK$, so $G$ is the internal Zappa-Szép product of $H$ and $K$, but neither $H$ nor $K$ is normal in $G$, so this is not equal to a semidirect product.


