Launch the Facebook app from UIActivityViewController instead of the default share sheet.
The goal and some implementation details are discussed in my blog post "Hacking the default Facebook UIActivity".
You'll need to install the Facebook iOS SDK.
If you are a Cocoapods user, just use pod install in the Sample directory.
You can also install the Facebook framework all by yourself and drop it into the project.
You will also need a Facebook app ID / display name to configure the project's Info.plist.