It is a little like a sick feeling, after a washing cycle, to start to pull your clothes out, only to find them dripping wet with water, sitting on the bottom of the wash tub. The error codes says the shifter/actuator is bad. Don't believe it. Clear the error codes as per page 3 of the service manual, then run the automatic test on page 4 of the service manual, this is what worked for me.
Kenmore 110.20022013 Service Manual 110.20022013.pdf
Update: December 11, 2019. The problem was this, the splutch, that the shifter/actuator pushes back and forth, was very stiff and hard to move, by hand, I took it apart, and the plastic parts that moves one on another, was clean, had no grease or oil. I oiled the splutch and part it moves in, using what I had on hand, 3-in-One oil. That worked as it was easy to move the splutch after this. Now the shift/actuator had an easy job pushing the splutch back and forth changing cycles. I did a load of clothes yesterday, worked great!
Update: September 12, 2019 - W10006355 shifter/actuator (don't buy until clearing the error codes and then running the tachometer test and then the automatic test, to be sure it is bad or good)
I found on applianceblog that you must clear the error codes(page
3 of the service manual),
when you replace the actuator/shifter
W10006355.
I didn't have a new one, so I used an old one I had, that I thought was bad,
and amazingly, after clearing the error codes, it worked, ran the
automatic test, then it did a load of clothes, agitating and spinning dry,
sticking to the sides of the tub like normal.
I have 4 shifter/actuators-W10006355
I bought, over the months/years, trying to fix the
F7E5 error code. One I fouled up soldering on the circuit board. The
other 3 all work, I found out after running the tachometer test-p-5 and the automatic test-p-4. These were all thought to be bad because of the F7E5 error
code. All I really needed to do was to clear the error codes as described on
page 3 of the service manual. I didn't need to purchase these shifter/actuators
- W10006355,
just needed to clear the error codes.
The computer chip is a "STMicroelectronics STM8S207 RBT6" Mainstream Performance line 8-bit MCU (Microcontroller) with 128 Kbytes Flash proram memory, 24 MHz CPU, integrated 2 Kbyte Data EEPROM, 6 Kbytes RAM, 64-PIN LQFP.
These Kenmore models share the same service manual they call a "Tech-Sheet". It contains diagnostics codes and how to trouble shoot the machine to find the defective part.
110.21302013
110.21392013
110.21252112
110.21352112
110.20022013
110.21102013
110.21182013
110.5072011
110.21292013
110.20102310
110.25102310
110.23102310
110.21492310
110.20102311
110.20112310
I found the same printed service manual as the pdf above, taped to the front of the
machine underneath the inside at the top front. You need a long arm to reach
that high from underneath the washer while it is tilted backwards.
Kenmore 110.20022013 Parts List pdf file
Bought this Kenmore-Whirlpool washer in 2014, the only part that has broken is the Actuator-Shifter, (per diagnostic codes F7E5) Whirlpool part no. W10006355 or WPW10006355. About $37 to $40 dollars on ebay, or $65 local.
Have had 3 service calls, the first service call, I don't remember for sure, but it probably was the actuator-shifter, the second service call it was the actuator-shifter W10006355, the third service call it was the actuator-shifter W10006355, and the fourth time the washing machine broke, I replaced the actuator-shifter W10006355 myself. The sign or symptom that it is broke, is at the end of the wash cycle, the clothes are dripping wet with water. It might or might not agitate, and it doesn't spin.
It is very easy to replace. (But you most likely don;t need to, just clear the error codes on page 3 of the service manual) Unplug machine from electric. Tilt machine backwards, so you can work underneath it. Remove the clear plastic cover with two large 5/16" screws, then remove the actuator-shifter with two phillips head screws and a large connector plug with two tabs you have to depress as you unplug. When replacing the actuator-shifter W10006355, make sure to place the plastic pin of the actuator-shifter W10006355, back in the hole of the plastic splutch, so the actuator-shifter W10006355 will engage the splutch.
This VMW, verticle modular washer is very easy to replace the actuator-shifter W10006355.
Kenmore 110.20022013
Whirlpool part no. W10006355 Actuator-Shifter 
Great washing machine once you learn how to repair it
yourself. (by clearing the error codes)
The shifter/actuator
W10006355 has a "ON Semiconductor 14106BG Hex Schmitt Trigger SOIC 14-NB package surface mount
i.c.".
I suggest keeping the washing machine unplugged except while washing.
Sears help
https://www.searspartsdirect.com/washer-repair/symptom/WASH009.html
How to Read a Vertical Modular Washer (VMW) Error Code Display - video by Sears,
it tells how to access the fault codes and error codes so you can tell which
part is broken.
https://www.youtube.com/watch?v=VZoCjjguxRI
Whirlpool Vertical Modular Washer (VMW): Tech Sheet, Diagnostic Mode, Fault
Codes, and Mode Shifter - video by a Technician, he calls the actuator-shifter a
"mode shifter".
https://www.youtube.com/watch?v=0zmzSSUc8dI
Washer Actuator - buy at Sears
https://www.searspartsdirect.com/part-number/W10006355/0026/110.html?modelNumber=11020022013&categoryName=Residential%20Washers&brandName=KENMORE