Thank you for the extra information.
There are several things you can do to deter this behavior. She/he is marking her territory. This is common in all cats male/female fixed or not. Females are far less likely to do it, but are more than capable of it. Most people think that cats only pee to mark, but that is untrue. They will poop where they shouldn't, and also furniture scratching is a marking behavior too.
Try some of these tips to stop the behavior.
You should add a second litter box.
Many times cats do not like to urinate in the same box they defecate in. Nor do cats like to share litter boxes either sometimes. It always a good rule of thumb to follow this formula (n+1= number of litter boxes) N= number of cats in house.
You can use a litter called cat attract. Attract Litter is designed to make it easier to get cats of all ages trained or re-trained to use the litter box. You can get this a petsmart, petco, and larger pet supply stores.
Very important to clean the areas that has been marked. You need to use a enzyme odor eliminating cleaner such as nature's miracle. Regular cleaner such as Lysol will not work as good. She will keep going to the same spots because she smells her scent there. You must get rid of the scent. Nature's miracle also makes a cat only formula now.
You can also use a product called feliway. This is a pheromone spray that helps cats with aggression, territorial issues, and anxiety. You can spray it on surfaces she has previously pooped on. You can also get this in an electric plug in that puts the pheromone into the air. You can find this a Pet Smart and most larger pet supply stores.
When you do catch her using the litter box appropriately reward her with a yummy little kitty treat. Cats learn very fast a reward method of training.
If she starts to have abnormal stool or trouble going, you need to get her a check up. Cats will avoid the litter box if medical conditions arise that make going painful or uncomfortable.