It would help to know what you are really trying to do as there are likely solutions that you may not have thought of.
Like using an inexpensive door lock module instead of trying to use separate relays.
i.e. a small used car door lock module may be cheaper than the relays you will need to drive this.
Many door lock modules will also limit the time duration of the power to the solenoid so you won't have to worry about that either.
And some use the sensor wires to know when the actuator has moved.
(I am very familiar with the one from a 99-00 Mazda miata)
In terms of what you have. It is a typical 5 wire door lock actuator.
The blue and green drive the actuator. power one way moves it one way; power the other way moves it the other way.
The other 3 wires, black, white, brown, are sensor signal wires that connect to an internal switch that can be used to tell where the plunger is.
The black is common, the white and brown will indicate the position of the plunger.
One or the other will short to the black wire to indicate that the plunger is at that position (out/in).
If you google around for 5 wire door lock actuator / solenoid you should be able find more information.