You can only send event invitations to friends and group members, not to everyone that likes your page. However, you can post the event on your page, so that it will appear in the News Feed of people that like your page. The post will have a Join link for them to RSVP to the event. If you want more exposure than that there is also the option of purchasing a Facebook ad for the event, in which case numerous targeting options are available (including the ability to target only people that like your page, or friends of those people, or everyone within 50 miles of your city with a certain interest, etc.).
I use the same type of group for my Fraternity's page on Facebook. We need to let members know information that is going on, and some of which can't be seen by others (especially uninitiated members :D).
The following is a screenshot a setting under "Edit Group" (you can get to this by going to the group page, then clicking on the button at the top right of your page).
Click here to see.
What you need to do is either have a "Closed" or "Secret" group (which I currently have my group set at).
Open Group
"Anyone can see the group, who's in it, and what members post."
This is typically used by businesses and things of that nature that want to get the word out that, for example, there will be a sale on Saturday or a special on beer. Anyone can see it and there are no restrictions.
Closed Group
"Anyone can see the group and who's in it. Only members see posts."
This can be used by a business or an organization like what my fraternity is. Basically you are giving the people who are in the group access to the posts. BUT, different from "Secret" group, people can actually see the page in search (and can ask to be in the group).
Secret Group
"Only members see the group, who's in it, and what members post."
As mentioned above, this type of page does NOT show up in search because the administrator only wants those people in the group to be able to see the posts. This goes great for a fraternity, because certain things such as ritual should be secret to only those initiated.
Basically how I see this, if you only want people in the group to see posts, you have the option of "Closed" or "Secret". From there you can answer the questions:
(1) Do I want the group to show up in search?
(2) Do I want people to see the group from the outside, and be able to request in?
Hope this is helpful... if you have any other questions about groups you can go to facebook and they have many help and FAQ sections to help (I'm limited to two links in this post because I'm new).
Best Answer
Yes, it is possible to tag a Facebook group in a post. Below are instructions:
Step 1
Log into your Facebook account and then navigate to the item you'd like to update with a tag. Click on the "Comment" hyperlink beneath a status or image posting to tag a Facebook user in the post's comments sections. To tag a group to your status, click inside of your status box at the top of your news feed.
Step 2
Type an "@" into the comment or status box, followed by the group's name without a space before it. For example, type "@soccerclub" to tag a group named "Soccer Club" to your post. Facebook will attempt to predict the group for which you're searching. Click on the group's name when it appears beneath the comment or status box.
Step 3
Click on the "Post" button to submit your status update with the desired group tagged in it. For comments, hit the "Enter" key on your keyboard to post your comment and its tag.