Sharing to Facebook Groups and Facebook Event Pages other than your personal profile page is not a feature we offer at this time. You can always copy and paste your workout link to the Facebook Page to share it though!
Please be aware that sharing workouts and routes shares this information with everyone allowed to see your posts on Facebook, there is no option to turn this off. You can ensure your workout or route's complete privacy by editing your privacy and map privacy settings on the site. You can find out more about privacy settings here.
Please note: sharing imported workouts is not a feature that we offer at this time.