I am trying to solve this by Van-Kampen's theorem. What I do is just move a point at left side and a point at right side. Then I get two open sets whose intersection deformation retracts to a circle. Each open set deformation retracts to a figure 8.
So what is the next? Just free product with 4 basis module the normal subgroup generated by the fundamental group of a circle? I can not see what is this normal subgroup. Is it just Z?