Hi iods,
Below is an outline schematic of a square wave inverter to give you an idea of what would be involved.
spec
NOTES
(1) The above inverter would produce a square wave as opposed to a sine wave.
(2) The output would be a 220V peak square-wave (the same RMS value as a 220V RMS sine wave)
(3) The NMOSFET types are not shown. There is a wide range of NMOSFETs but some pretty hefty types would be required for a 500W RMS output. With a 500W load the current taken from the battery would be around 50 Amps.
(4) The NMOSFETs would require substantial heatsinks
(5) The NMOSFET driver chip does not provide dead-time. Dead-time would be advisable in practice. Dead-time is where there is a delay between one NMOSFET turning off and the other NMOSFET turning on. This avoids the situation where both NMOSFETS are conducting together, which is very undesirable, both for the NMOSFETs and the transformer.
(6) Capacitors are disc ceramic with an X7R dialectic.
(7) An LM324 operational amplifier would not be the best choice for a 50Hz oscillator. An LM393 comparator would be better but, an LM555 timer would be better still (the circuit would require modification to use a standard LM555 timer). And an LMC555 CMOS version- as shown- would be better again. But, if you really want to use an LM324 oscillator, just say and I will post a schematic for a suitable circuit to replace the LMC555.
(8) It would be possible to generate a precise, highly stable, 50 Hz frequency by replacing the timer chip with a crystal-controlled 50Hz generator. But that would be more complicated.
(9) The NMOSFET driver chip is necessary to drive the highly capacitive NMOSFET gates with high currents so that the NMOSFET turn on and off fast.
(10) The resistors in the gates of the NMOSFETs are for shaping the waveform at the NMOSFET gates.
(11) Snubbers, which would probably be required on the NMOSFET drains, are not shown.