I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).

Suppose we have a group $G$ and subgroups $A$ and $B$ such that $A \cap B = \{1\}$ and for every $g \in G$, there exists $a \in A, b \in B$ such that $g = ab$. Then we can express $G = A \bowtie B$ as a Zappa–Szép product. This of course reduces to the semidirect or direct product in the nice cases.

Then suppose, we have sufficiently nice representations of $A$ on an $F$-vector space V, and $B$ on an $F$-vector space W, then can we find a representation of $G$ which in some sense preserves the representations of $A$ and $B$?

I've been told that the solution for semidirect products uses something called Clifford Theory, but we don't have a semidirect product here.

Our problem involves a monoid, not a group, but the Zappa-Szep product is constructed the same way there.