I installed a cat door. It had a magnet in the bottom of the flapping part and another magnet (or metal piece) in the rim of the opening. Their mutual attraction held the flapping part firmly in place. It did not open without a firm push.
It may help if you were to substitute a smaller magnet, or smaller metal piece. Then your cats might have an easier time pushing the door.
Ideas for devices to open and close the door...
A servo (the type used in radio controlled craft) would be able to pull open the door. The control signal consists of pulses of particular lengths, to position the servo properly. Such a control circuit will take some electronics skill to construct and adjust.
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Motor operation:
A motor can pull the door open via a string. The motor might be a small hobby motor, or the kind that drives an audio cassette player.
A clutch mechanism should be installed on the shaft, lest the motor pull too strongly on the string. A clutch will take skill to make.
The motor might operate and stop after a few seconds via a 555 timer IC. Or it might be stopped by a circuit which senses a stalled motor.
To lower it you must run the motor in the opposite direction, which requires reversing the power supply. It is possible but again it will require some skill to construct the control circuit. There may be an IC to do this, but I don't know which number it goes by.
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Detectors:
It will take skill to construct the photo detector circuit, and to get it to trigger in the proper circumstance.
You probably would want to build two detectors. One for indoors, one for outside. A cat will interrupt one detector and activate the door opener. Then interrupt the second detector to close the door.
As a failsafe you may want to install a detector within the doorway, to detect that the cat is going through.
Or as an alternative, let the door open for 10 seconds, then close automatically. This will handle occasions where the cat triggers the door opening but does not go through.
The light detector can be a cadmium-sulfide photocell. You might illuminate it with an IR led, or a narrow beam led.