Why does Aunt Alexandra invite herself to stay over at the Finch house?What is the real reason? Atticus doesn't seem to really be that kind of person.
At the beginning of Chapter 13, Aunt Alexandra tells Scout that she and Atticus decided that it was time for Scout to have a feminine influence. Of course, Scout thinks to say that Calpurnia is a girl, but thought better of it and remained quiet. Aunt Alexandra fits right in with Maycomb and this is because she's part of that tradition of believing in tradition and heredity. She, like many people in Maycomb, believes in continuing social and familial trends; this and looking after Scout are the main reasons she comes to stay with them. She is not as stubborn as many of the other traditionalists in town, which could be Atticus' influence, but she definitely tries to teach Scout that, for instance, their family line is just plain better than the Cunningham's.
Atticus does tell Scout that he can't watch them all day; he says this in attempts to try and defend Alexandra's decision to come stay there. And I would assume that he was more than comfortable with Calpurnia playing the mothering role for Scout and Jem. Atticus also has no interest in social and familial traditions because he believes in the ability for society to change and because he doesn't want to be proud simply for the genetic accident that he happened to be born into a family that had more money than another.