On Zappa--Szép products of a wreathed $2$-group and a cyclic group

Let $n,m\ge 1$, and let $H=C_{2^n}\wr C_2$ be the wreathed $2$-group and $K=C_{2^m}=⟨z⟩$ a cyclic group. We classify the Zappa–Szép products $G=HK$ in which the base $B\cong C_{2^n}\times C_{2^n}$ of $H$ is normal in $G$. When the matrix $Z$ of the $z$-action on $B$ commutes with the swap $J$ of the two base generators – equivalently, $Z$ is symmetric circulant – we classify these products by an explicit system of seven polynomial congruences in a tuple $(p,q,r,s,c)$. Dropping this hypothesis, we obtain a unified classification of all such products with $B$ normal by five congruences on the entries of $Z$ and the parameters $(r,s,c)$, of which the symmetric case is the specialisation $JZ=ZJ$. Finally, we separately treat the classification for the modulus $M=2^m=4$, since in this case the congruences degenerate.