  We investigate the $\alpha$-colored Eulerian polynomials and a notion of
descents introduced in a recent paper of Hedmark and show that such polynomials
can be computed as a polynomial encoding descents computed over a quotient of
the wreath product $\mathbb{Z}_\alpha\wr\mathfrak{S}_n$. Moreover, we consider
the flag descent statistic computed over this same quotient and find that the
flag Eulerian polynomial remains palindromic. We prove that the flag descent
polynomial is palindromic over this same quotient by giving a combinatorial
proof that the flag descent statistic is symmetrically distributed over the
collection of colored permutations with fixed last color by way of a new
combinatorial tool, the colored winding number of a colored permutation. We
conclude with some conjectures, observations, and open questions.
