WARNING: this is a human activated sequence and as such is prone to ERROR. If you activate refill before tank light is on, you may get a big puddle on the counter and the floor! A water sensor can be added to detect over fill and shut off the water valve.
First, I installed a tube that hangs on top of the tank in a way that it does not obstruct tank in/out path. The tube comes out through one of the existing holes at the bottom. Here is how:
Refill in action. Note that I am restricting the flow with a valve as otherwise it is too strong and water splashes all around.
These are the pats involved: blue tube is connected to the house water supply. Water then goes through a valve (partially open in order to restrict the flow a bit), and then through the solenoid that controls on/of flow. This is a 24DC normally closed solenoid so I got a 24DC power adapter (1A). The adapter is connected to a smart plug that one can program and integrate wit Google Assistant or Alexa. All parts were purchased from Amazon with total cost around $40.
A few more details. I restricted the flow so that tank fills to about 4/5th of capacity in 1.5 minutes and then created a program (a scene) sequence for the socket that controls the solenoid and gave it a name, "Refill" in my case. The sequence does following: (ON, 1.5 minutes timeout, OFF).
I have used this for three refills already I found it very satisfying. I will still have to pull out the tank once a month or so to recharge/change the water softener filter and to clean the tank but that is quite manageable.