We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to "previously unknown" periodic orbits in the planar problem.