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Let $\{G_n\}_1^{\infty}$ be a sequence of non-trivial finite groups. In thispaper, we study the properties of a random walk on the complete monomial group$G_n\wr S_n$ generated by the elements of the form$(\text{e},\dots,\text{e},g;\text{id})$ and$(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\inG_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on$G_n\wr S_n$. We find the spectrum of the transition probability matrix forthis shuffle. We prove that the mixing time for this shuffle is $O\left(n\logn+\frac{1}{2}n\log (|G_n|-1)\right)$. We show that this shuffle exhibits$\ell^2$-cutoff at $n\log n+\frac{1}{2}n\log (|G_n|-1)$ and total variationcutoff at $n\log n$.

Cutoff phenomenon for the warp-transpose top with random shuffle