Quoted from chas10e:

the wrapper does looks a little toasty but the coil may be ok ( when removed check with an ohm meter)
The coil sleeve is probably what pieces are falling out, order a new one of those as well
You will want to clean the whole stepper unit so things work smoothly
Part catalog here: http://www.planetarypinball.com/reference/partsmanuals/BLY_Parts_1976/index.html#/106/
Some great info here about working on EM games: http://www.pinrepair.com/em/
I have ordered from Steve Young at The Pinball Resource: http://www.pbresource.com/coils.html can call him on the phone and ask for the correct coil sleeve as well as the coil part # given above.

Quoted from MarkG:

Is this for your Bally Champ? Either way, the details you need are often in the schematic:
[quoted image]
Note that they're not the same solenoid. Based on the wire colors in your photo I'd say that the coil shown is the Bonus Unit Step Up solenoid, which looks like B-26-1100.
The reason they're not the same is that stepping up a stepper usually takes more effort than resetting since in addition to moving the stepper it's also winding up the spring that will allow the stepper to step backward or reset when the reset solenoid fires. The reset solenoid usually just has to move arms out of the way of the gear so the spring can unwind and do the work of moving the gear back.
In this case both solenoids have 1100 turns of wire, but the Step Up solenoid uses 26 gauge wire which allows more current to flow and delivers more power than the 28 gauge wire used by the Reset solenoid.
/Mark