If Izzy is not going potty outside then he does not know that is what he is supposed to do. These setbacks happen often in puppies (bigger ones will come around sexual maturity around 6 months) and you just have to go back to the basics and reteach.
Every time he has an accident inside he builds that habit. You need to make sure you do all you can to keep him from developing a habit you do not like. This means LOTS of supervision, crating when he can't be watched, and whisking him outside at the slightest hint of having to go. Set him up for success and you can't fail!
With the playing outside then eliminating inside, I would recommend remedial crate training. Take him outside but keep him on leash. BUSINESS BEFORE PLEASURE.
No playing, walks, or fun until he eliminates. Wait only 5-10 minutes then take him back inside if he does not go potty. Carry him directly to his crate (so that he cannot have an accident) and leave him there for about 10 minutes then try again. Keep repeating this process until he DOES go outside (he should have no opportunity to go inside so he is set up to eventually succeed.) Throw a PARTY! Lots of praise, treats, toys, walks, fun. All good things happen when you eliminate outside. Then you can happily invite him into the house and let him be out with you.
Keep in mind that within 30 minutes after eating, drinking, waking from a nap, or exciting play his body will start telling him that he needs to go to the bathroom so the process will start all over again. Pretty soon he will learn to get it over with quickly and life will be much easier
With cold or wet weather you can try booties to protect his feet from the wet and cold, hold an umbrella over him, find an area with very short grass or rocks that he can go on so that he doesn't need to wade through the long wet grass. Little dogs HATE getting wet or being cold, you can try these things to make it easier but in the end Izzy just needs to be able to go to the bathroom no matter what. Tough love, follow the previous protocol of outside-crate-outside-crate until he goes. Remember to reward heavily when he does go to make it worth his while.
Hang in there, the more work you do not the easier it will be in the long run!