F-droid (0.7) has two options:
One to grant it "root" powers, and another to give it "system-app" powers.
Just using it as a "root" app will let it install apps without going through the whole "Are you sure you want to install this app" page.
What does System-app give it?
I can't test it as it crashes on my phone
I believe this was/is a planned feature for F-Droid, which would essentially give it features similar to the play store where it can auto-update apps, install them and remove them all by itself, without needing user interaction. See this as an explanation - it's the extension which was meant to give F_Droid these priviledges, but as it says in the description:
Note: This is still in a testing phase. Current F-Droid versions do not yet interact with this extension.
Information from the F-Droid Site:
This is for an app called System App Mover, but it's description is correct.
System apps can get more privileges, so some apps get more functionality when installed as a system app.
On the other hand, system apps can not be uninstalled. So this app can also be used to convert system apps to normal user apps by moving them from the /system/app directory to /data/app directory.
WARNING: Uninstalling important system apps might result in a unusable device! Use this function at your own risk and only if you know what you're doing!
See What do android application permissions mean. There are some permissions that have a "level" of "system". Those can only be used by applications that are "system" applications. I do not know why f-droid would need them. Maybe the INSTALL_PACKAGES permission. This would allow f-driod to install an application, like Google Play does, instead of sending the Install request off to the Package Manager application.
