Let me set some context, before I explain. My site is a 19 year old virtual club for enthusiasts and collectors of a specific brand of antique garden tractors. It includes a significant online library of contributed resources of all types, plus an active Classifieds ads (free) that allows them to buy, sell or trade for free with real folks avoiding Craigslist scammers and hassles ( we have always required users to provide their real name, city, state) or eBay costs. It also supports an aftermarket of reproduction parts. In addition to researching and finding things or asking questions, users can document their collection and their projects in a variety of ways (a custom tractor registry - now using Pages + collections, photo gallery, build log blog, a Show & Tell forum specifically for that, etc.)
We also have a Facebook page and a Facebook group. The page is simply to stake a place and hang a shingle on Facebook - because we are a virtual club, many of the FB tools for a business owner to use on their page simply does not apply - and most imprtantly, because search engines and such penalize me if I don't have one. Consequently that page doesn't see much traffic, and the content is not very dynamic. The Facebook Group is used to "push" key things from my site into Facebook so that users can then share that info with their friends, family, other groups, etc. We share new classified ads, blog updates, new photo gallery items, new registry entries, etc - so that our users who are in Facebook can then share their hobby with their friends, family, etc. My site has over 9000 users, and the FB group has over 1000 members.
We do not actively encourage users to post in that FB group, much less use it for general chit-chat. Similarly, we don't encourage status updates on the club site. I really don't care what they had for lunch, what their kids/grandkids now look like, or where they went on vacation. But, if they have something significant or meaningful related to their tractor hobby, I want to capture that, catalog/index it, and save it for posterity (rather than it disappear with the latest Photobucket policy change, when MySpace or or other free photo repository fades away, etc.) I want to add that to our knowledge base and repository and make it easily searchable and shareable.
In addition, we have a News and Announcements forum for news about the club, the site, or its software - and we share all topics in it to Facebook. Meanwhile, our moderators also monitor that Facebook group and should someone ask a question there, they respond and if possible provide a link back to the site for more details, resources, etc.
We try to use Facebook as a means for our users to share info about their hobby with a wider audience, to "market" who we are and what we can provide to a wider audience (especially younger users who live in their phones but may seldom use a computer), and as a way to broadcast news and updates about the site and the club. I don't try to compete head to head with FB, while fully acknowledging that it is a part of many of our users lives, and simply want to facilitate that...
So, things I would like to see are:
1. Ability to Promote to a Group in addition to a Page
2. Ability to auto-promote new content from specific parts of my site, to reduce the workload of moderators manually sharing (or promoting) content, and let them work in a "by exception" mode, similar to how they police the forums.
3. Ideally, I'd like to see the ability to automatically feed or pull back to my site any Facebook reactions or comments to content that has been shared or promoted there...