Dan, door darters like Oscar can really be worrisome as we all know what can happen to them out there in the world of mean people, predators, and traffic. Some people simply try to distract their cat when the door is opened, or put them in another room. We feel very strongly that it's possible to train a cat to observe this boundary through positive reinforcement. Of course, it will be much clearer to Oscar if he's never allowed outside at all. It'll be much easier for him to understand consistent rules that everyone in the household enforces.
It's important to understand why Oscar might want to run outside. Many indoor cats are bored with their surroundings and see a lot of cool stuff happening on the other side of the window. You can help by making Oscar's indoor environment more interesting. This doesn't have to be expensive. If you get a package in the mail, cut the box flaps off and share the box with Oscar. When you come home with groceries, leave a paper bag out for Oscar to sniff and explore. Take every opportunity to add temporary playthings to Oscar's world. Smells are like stories to cats and bringing in interesting scents from the outside world can be highly stimulating.
The next step is to make the doorway less attractive to Oscar. Please note that cats do not respond well to negative reinforcement, so punishing Oscar in any way will only communicate to him that he was right to try and get away from you and your home. Instead, you want to get a good supply of his favorite treats ready. You're going to need them.
A lot of cats rush to the doorway when their humans leave or come home because they receive generous amounts of attention then. We want to change that by moving the area where Oscar gets greeted and given his farewells. We want him to associate positive things with that new spot in place of the doorway. It can be a bed, a cat tree, a mat on the floor, or any clearly defined area.
Begin by calling Oscar over to the spot where you'd like him to be. When he gets there, offer him a treat and pet him. Once he's wandered away again, call him back and reward him again. Do this a few times each day until he eagerly runs to the spot you've chosen when you call him there. You can even give the spot a special name so he'll understand what you want. If he isn't interested, let it go for the time being and try again later. You want this to be a fun interaction for Oscar, not a forced training session.
Now, repeat this behavior each time you leave the house and each time you come home. Oscar should soon learn that being in the right place at the right time earns him attention. It also shows him that when the door opens, being at the door earns him nothing but a set of grabby hands intent on keeping him inside.
If his behavior persists (some cats can be very stubborn about this), a deterrent can be used. Our preference is for a citrus sachet made of fresh orange peels in a mesh bag that can be hung from the doorknob. One whiff of that should have Oscar recoiling from the door without blaming any humans for the offense.
Good luck, Dan. We wish you and Oscar all the best!