Homogeneity degree of cones

The homogeneity degree of a space X is the number of orbits for the action of the autohomeomorphisms group of X. We determine the homogeneity degree of the cone over a locally connected curve X not being a local dendrite in terms of that of X. Using the result of Pellicer-Covarrubias and Santiago-Santos, it gives us a formula for the homogeneity degree of the cone over any locally connected curve X.