  We study the components of the Hurwitz scheme of ramified coverings of
$\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements
in the conjugacy class of product of two disjoint cycles. In order to detect
the connected components of the Hurwitz scheme, inspired by the case of the
spin structures studied by Fried for the $3$-cycles, we use as invariant the
lifting to the Valentiner group, triple covering of $A_6$. We prove that the
Hurwitz scheme has two irreducible components when the genus of the covering is
greater than zero, in accordance with the asymptotic solution found by
Bogomolov and Kulikov.
