Group Actions · Hard problem

Label the vertices of a regular decagon $0,1,\dots,9$. Let $C_{10}$ (rotations only, no reflections) act on the $5$ long diagonals $\{0,5\},\{1,6\},\{2,7\},\{3,8\},\{4,9\}$ joining opposite vertices. Find the order of the stabilizer of $\{0,5\}$.