On Jacobians with group action and coverings

Let $S$ be a compact Riemann surface and let $H$ be a finite group. It is known that if $H$ acts on $S$ then there is a $H$-equivariant isogeny decomposition of the Jacobian variety $JS$ of $S,$ called the group algebra decomposition of $JS$ with respect to $H\mathrm{.}$ If $S_1\to S_2$ is a regular covering map, then it is also known that the group algebra decomposition of $J{S}_1$ induces an isogeny decomposition of $J{S}_2\mathrm{.}$ In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.