Welcome to JustAnswer. I'm a licensed veterinarian and I'll be happy to help. There are three main concerns or areas that you need to address. First of all, If Abba isn't already neutered he should be right away. Female cats are in heat right now, so this could be what is drawing him away.
Secondly (and it sounds like you may have addressed this ), don't leave any food
out for Abba after his breakfast, and then offer him a delicious dinner when you want him to come back in. Canned cat food is often used for such enticement, or you could even give him a little treat of chicken, steak, fish, etc. before his main meal.
Finally, it would be good to enrich his indoor environment with a cat tree, toys, and play with you.
I do really wonder with how you describe his behavior if he isn't neutered.
Look forward to hearing back from you,