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

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

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Compressors, Gas Turbines and Jet Engines
In the cross compounding of the gas turbine plant

H.P. compressor is connected to H.P. turbine and L.P. compressor to L.P. turbine
Both the arrangements can be employed
All are connected in series
H.P. compressor is connected to L.P. turbine and L.P. compressor is connected to H.P. turbine

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