I Square waves enter the input terminal and output amplifies them. But the first mosfet part of the circuit is probably a push-pull circuit. However, I don't understand what role the MOSFET part in the back plays. Also I've tried making this LTSPICE but it doesn't amplify and Gate comes out weird and I don't know what's wrong. I would really appreciate it if you could let me know this.

 

Try putting your inputs out of phase with each other.

 

Q4 seems to connected with drain and source reversed so the body diode will not allow the sourcepins of Q3 and Q4 can never get positive of V-.

 

It gives the impression of being a driver for high powered mosfets., where the second stage either supplies the switch on charge or removes the switch on charge. So the power devices being driven are probably in parallel.
What can the TS tell us about the physical part of the system, Is any context available??