Lifts of simple curves in finite regular coverings of closed surfaces

Suppose \(S\) is a closed orientable surface and \(\tilde{S}\) is a finite sheeted regular cover of \(S\). The following question was posed by Juli\'{e}n March\'{e} in Mathoverflow: Do the lifts of simple curves from \(S\) generate \(H_{1}(\tilde{S},\mathbb{Z})\)? A family of examples is given for which the answer is "no".