Liz's Mom. (Spoilers)
Liz is a FBI special agent and has no doubts about this lady that just came into her life? She knows her mother just kidnapped Red and could be coming after her. Her neighbor moves out unexpectedly. This new neighbor fits her mother's age. Wouldn't they have asked Red to give them a description of her and have a FBI artist draw a picture of her? At the least Liz would have asked Red what she looks like.
I love the show but this seems far-fetched. Liz has this lady watching her daughter and she never checks into her?