There is a fun thing called an AgiTamer that is a rubber-like circular cuff (with a small section missing) that you tug and pull under the agitator skirt. Some water and liquid detergent help, although you then have to wear good, grippy gloves to pull the Agitamer as far around the agitator as possible and tuck up under the skirt. Then you screw the little attached hose to the water faucet and SLOWLY turn on the tap, WITH THE WASHER LID HELD CLOSED. The agitator might come up just a little, in which case you turn off the water and release the pressure in the cuff so that it can be tugged further around and under the skirt. Then you start over. The agitator will either fly up or the cuff will explode. In either case, you want the lid as closed as possible. Old Bakelite Gyrators will sometimes separate from the metal spline, leaving it on the agitate shaft when the Gyrator pops loose.
Before you start with the process, you will need to heat enough water to boiling to pour into the center of the agitator onto the hub so that the hub at the bottom of the agitator barrel is under water. This is going to take the capacity of all of your large pans and you should strive to have the whole hub surrounded by water pretty fast, otherwise the heat from the water is going to escape and just heat the tubs. If you cannot come up with that much boiling water, you can try pushing a heavy but flexible bag down into the agitator so that you can pour boiling water into the bag in the agitator barrel and the plastic bag will relax enough to cover the top and sides of the hub to transmit heat into it. If you use the plastic bag, put a blanket or something similar over and around the agitator to help hold in the heat. Then close the lid and let the heat work for 15 minutes to half an hour. Drain the water and then try tugging. Put soft towels or something similar in the agitator and have the soft, thick fabric hanging out around the top because you are going to have to have your face close to the top to be able to try to lift from near the bottom. If it suddenly lets loose, you won't have injuries on your face. If possible, try to lift by the barrel and not by the skirt. It is more efficient and safer. If this is a PowerFin, you can get your fingers under the openings where it joins the skirt. If tugging does not do it, then use the Agitamer.
When you replace the agitator, use lubrication on the agitate shaft and you won't have this problem again.
