On Zappa--Sz\'ep 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 \(K = C_{2^m}=\langle z \rangle\) 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.