Betty, the legitimate daughter of Rev. Parris, is likely frustrated with the attention she's had to share as Abigail, an orphan cousin who has come to live with them. Abigail is demanding of negative attention and has likely taken the spotlight comepletely off of Betty.
Betty probably also wants to please Abigail at times. This is sibling rivalry come quickly. Half the time, I think Betty would be out to get her, but at other times she's happy to join in with the other girls' activities and feel that acceptance of belonging to a group.
Also, Betty is young... 5 I think. She is probably scared to death of what they are doing, but can't say anything.
You have this tagged with Act III, but I am going to talk about Act I because I do not think you really mean Act III. Betty hardly does anything in Act III -- only stuff with the rest of the girls, accusing Mary.,
To me, Betty's motivation in Act I is to keep herself from getting in trouble. Her father is the minister and is a harsh man. If she admits that she has been messing around with charms and such, he will be really angry at her. So I think she is faking her problems so that she can claim she has been bewitched and everything is not her fault.