Purpose of Cap to Ground in this Circuit?
Having C2 there is identical to just making C1 = 2uF. The schematic creator doesn't know what they are doing and seems to think C2 is somehow different than C1 and has drawn it accordingly.
The only reason I can think of is if they are trying to use it as a snubber to suppress arcing across the button and so wire a capacitor directly across the button but that could cause problems without a series resistance. But that would depend on physical wiring given the lack of inductance and high resistance of the circuit it doesn't make much sense.