Two parts to the problem. First, you need to open the device and figure out if the buttons are mechanically latching, or if they are just a momentary contact, normally open type of switch. Check how many poles there are on the switches. Use a voltmeter to measure the voltage at both sides of the switch contact(s), measure relative to the ground wire that ties to the car.
Second, you need a circuit to generate the appropriate delays, and to simulate the switching action of what the switches do. You can't do this till you understand what is happening in One, above.