I need to find an irregular covering of the 2-holed torus. If a covering is regular, then $p_*(\pi_1(E,e_0))$ is a normal subgroup of $\pi_1(S_2,b_0)$, where $E$ denotes the covering space. So an irregular covering would not have this property. I was thinking if $E=S_2$ (which is just itself) is such a covering, but I don't know why or how to justify it. If the trivial covering is regular, then what would be an (easily understood) irregular covering of $S_2$? Thanks!