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”.