Your photo shows a copper band around one end of the coil intended to reduce buzzing. There should be a brass washer on the other end, between the coil and the frame, with the brass bolt passing through it. It's intended to keep the coil off the frame and to keep the frame from getting magnetized.
Quoted from ChipS:the bulb would pulse on and off the correct number of times, but the coil continued pulling continuously, rather than pulsing along with the bulb.
If the coil stays activated once the power is cut the circuit is fine but there is either a mechanical or magnetic reason the armature won't relax. With the power to the coil cut the spring should snap the armature back to its rest position.
If you can't find a mechanical reason for the binding (switch out of alignment, armature binding on its post, etc.) the armature and frame may be magnetized.
As an experiment you can see if putting a piece of paper or two (or tape) between the coil and armature changes anything. If the frame isn't overly magnetized the thickness of the paper can sometimes keep the armature far enough away from coil to let it work properly.
Ultimately you may want to read up on demagnetizing techniques or look for a replacement frame and/or armature.