Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

The compression in both the cylinders is polytropic
There is no pressure drop in the inter-cooler
The suction and delivery of air takes place at constant pressure
All of the listed here

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Compressors, Gas Turbines and Jet Engines
In a turbo jet engine, subsequent to heat addition to compressed air, to get the power output, the working substance is expanded in

Exit nozzle, which is a constant volume process
Turbine blades, which is essentially an isentropic process
Exit nozzle, which is essentially an isentropic process
Turbine blades, which is a constant volume process

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