Can a nursing home or my sister who has POA keep my mom from going to church with me?

Most nursing homes have some kind of service right there in the facility. Can you attend that with her instead? Can you come share a meal with her instead of taking her out? Can you take walks on the grounds or even in the hallways? Sometimes it's better not to take them from their safe place. Sorry. 

rabbittking, with dementia, your Mom could give the appearance that she is enjoying herself with the outside visits, but once returned to the nursing home she could be hard to handle.   This is not uncommon for the patient to become really confused after the fact.   Some patients would go to their rooms and pack their clothes saying they are moving out.  Or refuse dinner, or refuse their medicine.

Thus, keeping Mom within her safe cocoon of the nursing home would be in her best interest.

What does the nursing home and/or the POA do to prevent your mom from going to church with you and going to your home?

Has the POA given the nursing home orders not to let your mom leave with you?