I need to do when the module have connected power supply, it will automatically turn on the module. Because is there a PWRKEY button, probably grounding it or give on that pin VDD_EXT will not work. I've read whole Hardware documentation, but I found only schemes where is turning up via transistor or button, but nothing about automatically turn on. This feature have for sample SIM800L coreboard, but scheme of that board isn't available.
I have come up with my own discrete circuitry to generate the small negative going pulse after Power on. the green one is the final regulated power supply (3.3 V), The blue pulse will be initially high, goes low after about 10s of ms. And stays low for more than a second (requirement for Power button). then stays high for the whole duration of the operation.
you can tweak the values of C1 and R1 to vary the negative pulse width. C2 can be varied to smooth the rising edge of the pulse.
You can also omit R4 as there is a 100k Pullup already internal to module.
To automate the startup of sim800/sim900 module just keep the pwrkey shorted to ground always. As the supply will be provided modem will be on.
I've just been reading through the SIM5360 datasheet and it points out that you can just connect PWRKEY to GND to ensure automatic startup of the module (SIM5360_Hardware_Design_V1.06.pdf, page 28), and I think it's very likely applicable to the SIM800.
You can connect the the PWRKEY to ground through a resistor and then command the module with an MCU GPIO port. Make sure to keep the GPIO LOW all the time, specially during startup so that the module turns ON. To turn it off issue the AT commant AT+CPOWD=0, tu turn it back on, send a 1 second high, followed by 2 seconds low and then 1 second high.
This way you can turn it on / off at will and programatically.