Add a Button to Facebook Group

Hello, I appreciate your review of this project and your possible bid, but unfortunately Freelancer does not allow more precise budget limit categories. It is not a demanding job request, therefore I am willing to pay up to $50.00 for this single feature outlined below and in attached screenshots. Thank you. I have a Facebook Group and a Facebook Page. They are linked so that I as an individual admin of the group can perform all control and modification actions, and my Page is admin of the group as well, but the difference is that my individual profile can see & use the “Message” button that appears next the names of all group members, but when I manage the fb group as my FB Page as admin, the “Message” button disappears and I cannot contact the member in a private chat with FB Page profile. However, my FB Page has a “Message” button that allows it to have private chats with anyone who visits my FB Page. Then, another thing is that when inside the FB Group with my FB Page as admin, any member who places their mouse over that admin name or icon image, they are shown the “Message” button that if they click it, they would be creating a private chat over inside the FB Page. So, the FB Group members can initiate a PM to my group admin FB Page, but my FB Page admin cannot initiate to the group members. Also, anyone who visits my FB Page, can place a Comment on any Post, and when they do, there appears a “Message” button next to their name where we can initiate a private chat to individual fb profiles as my admin profile and icon. So, the thing I need, is a Facebook App that allows my FB Page to initiate private chat with group members as my FB Page admin. Thank you. P.S. The attached screenshots have some notes on them that you should also read.