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. If $$S_1 \rightarrow S_2$$ is a regular covering map, then it is also known that the group algebra decomposition of $$JS_1$$ induces an isogeny decomposition of $$JS_2.$$ 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.