House breaking older dogs

Question
We have 2 rescued dogs one a terrier mix male and the other a schnauser mix female, both fixed and both about 12-16 months old. The terrier will poop and pee on the living room carpet several times a week but the schnauser will do her business outside.  They have access outside through pet doors when we are home in the evening and night. They wake up my wife about 5 am to go out (she lets them sleep with us) but instead of doing his business outside the terrier will sneak back inside to go on the carpet.  They enjoy being in a crate during the day and quickly pee outside when let out. It's the morning when we have issues.  Thank you in advance for any help you can provide.

Answer
The problem with dog doors is they do nothing to housebreak a dog. You need to start from square one with this dog and pretend that you brought home a new puppy. Put him in a crate when he cannot be watched, and let him outside after he wakes, eats, and plays. Go outside with him so that he can be praised when he eliminates in the appropriate place. The last is extremely important as it will help to shape the behavior that is desired.

If the dog is only doing this first thing in the morning, someone needs to stand there and WATCH to see that this dog has emptied himself outside before letting him back inside, and if he cannot be watched, he needs to be put in a crate immediately upon coming back inside.