The aim of this work is to numerically investigate the nature of the equilibrium points of the axisymmetric five-body problem. Specifically, we consider two cases regarding the convex or concave configuration of the four primary bodies. The specific configuration of the primaries depends on two angle parameters. Combining numerical methods with systematic and rigorous analysis, we reveal how the angle parameters affect not only the relative positions of the equilibrium points but also their linear stability. Our computations reveal that linearly stable equilibria exist in all possible central configurations of the primaries, thus improving and also correcting the findings of previous similar works.