"She brought me on board with this new English group" - You could rephrase the sentence like this: She included me in this new English group." So, to BRING someone on board means to include them. However, to BE on board WITH something means that you agree, e.g., Simone: I think all people should be treated with respect. George: I'm on board with that.

"On Board" means that you joined someone. To "on board" someone lends an image to someone boarding a ship, so if someone literally brought you "on board" a ship, they had you join them. Therefore, when someone brings you "on board" at work, that means that they asked you to join, or you asked to join, and you did. A good synonym would be "She brought me along with this new English group"

To be "on board" can mean "to agree with something", "to participate in something", or to be physically on a sailing vessel (any type of boat) or aircraft (any type). To "bring on board" means "to add", "to include", and sometimes even "to invite" (depends upon context). Usually it means to add a person to a project or group for the purpose of participation. In your example, to be "on board with this new English group" means that the person included you in this new group so that you can participate in some way. Since she "brought" you on board, she may have even invited you, giving you the option to accept or decline. As for synonyms for this idiom, it depends upon context. In your case, you could substitute the idiom with "included" or "added". Example #1: "She included me in this new English group." Example #2: "She added me to this new English group." Besides being used in a statement, it may additionally be used in a question. Example: "Are you on board with this new English group?" In this case, it is asking if you are going to participate in the group. Again, depending upon context and the speaker's intention, it could be asking if you are in agreement with something about the group.