If it had Push Start, it must have some sort of Smart Entry System.
Usually, thieves have a device that can track signals from your key, as the smart key is always emitting signals, thieves can walk around your house until the the signal is detected in the device, then the device duplicates the same signal that was coming from your Smart Key, and then they will head to your car and your car will be tricked into believing that the owner/key is nearby, it will automatically unlock and thus granting access to your vehicle.