| Sound like the pump primes and the starter soleniod energises, I would check the starter motor connections disconnect the start motor cable clean it up and reconnect use a contact grease or silicon to cover the connection after assembly.
I have had a problem in the past when the starter carbon brush contact with the commutator was poor and I had to strip and clean the starter motor.
Or alternatively and more likely although the battery is showing charged it is not holding much of a charge therefore it energizes the starter solenoid but cannot turn the starter motor. Try using a booster or jump it using the car battery.
Just a few thoughts.
Regards | 
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