Well from my previous experience this could be happening for a number of reasons:
1. Sometimes when there are two cats leaving in the same household one of them can feel threatened by the other. In this case the best thing you can do is get a second litter box and observe the cat's behaviors with one another. However if you've had both cats for a long time this is not very likely.
2. Cats can get quite attached to their household and bringing them in a new envrironment can stress them out. There might be something in your house that makes your cat feel uncomfortable.
3. Maybe the issue is with your litter brand. Try changing the brand entirely, switch the litter texture and smell. Another chance is that there is not enough litter in the box.
4. The worst case scenario (and what happened to my cat) is that your cat is sick. According to the vet, my cat avoided using the litter box because she didn't want her infected feces in the same place she usually uses. It could also have a urine infection. In this case the only thing you can do is get her checked out.
I definitely recommend getting advise from a veterinarian you trust, and perhaps a pet shop owner as well. Change the litter brand, clean it regurarly and make sure there is enough litter in the box. Also if your cat has a regular spot on the floor where it poops constantly clean that area very well. Sometimes a cat will go to the same spot because of the previous smell. Scoop up any litter that is on the floor and clean the area with something not containing ammonia. You can also try moving the litter box to a different location in the house (perhaps the location it constantly uses if there is one). Something a bit more risky (you can create a mess) is monitoring your cat and if it shows signs it will poo outside of the box you can pick the cat up and place it in the box. A covered box would also be a good idea, as it will prevent your cat from throwing litter outside of the box (as you mentioned it does and then poops). Don't scold your cat when it doesn't use the litter box and instead reward it with a treat if it does.
But overall, first thing I would do is get the cat to a vet for a check up and some advice. I hope I helped :)
