You can certainly write Automations to do this, but you can’t work directly with groups. And you can either have them turn on together, turn off together or “mirror“ each other so that you only need one automation per trigger. But you have to have one automation for each trigger switch.
So when A turns on or off, have B and C perform the same action.
When B turns on or off, have A and C perform the same action.
When C turns on or off, have A and B perform the same action.
Alternatively, you can use a scene, but you will need two scenes: one for all on and one for all off.
When A or B or C turns on, activate the all on scene.
When A or B or C turns off, activate the all off scene.
Obviously neither of these is very elegant, but they will work.