In the Wikipedia article
Zappa–Szép product
, the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid out. Within that article lies a section entitled "External Zappa–Szép products" which details how to take groups $H$ and $K$ (when the groups meet certain properties and when certain functions $H\times K\rightarrow H$ and $H\times K\rightarrow K$ are defined) and **define** a multiplication on the Cartesian product $H\times K$, resulting in a group which can be seen to be the internal knit product of subgroups $H\times 1$ and $1\times K$, thus constructing the external knit product. The external version, though, looks to be rather unwieldy to use in practice, and this brings me to my question.

*Was a group ever discovered via the external knit product construction, that otherwise was previously unknown and where neither subgroup was normal?*

It is quite likely a real pain to construct the left and right actions (i.e., the aforementioned functions $H\times K\rightarrow *$) and then show all caveats are met, without *a priori* having a group there to serve as the roadmap. Hence, if this actually has happened, say to construct a counterexample to a given conjecture, then studying how the conjecture gave rise to the actions would be (for me, at least) a fascinating study.

As an aside, there are plenty of examples wherein a group factors as an internal knit product of subgroups, neither of which is normal. In fact, given an arbitrary group $G$, a Frattini argument applied to the right regular action of $G$ on itself provides a knit product factorization of the symmetric group ${\rm Sym}(G)$ as $G$ times ${\rm Sym}(G\smallsetminus 1)$. Thus, all but the smallest such groups $G$ provide the specified factorizations.