Even if such notifications look "official", they rarely are. They are specifically styled in a way to make them appear such, to betray people thinking that and tempt them to install the "stuff" behind. But one is more likely to install some malware that way, often a Trojan – like in your case: Its permanent switching on of network functionality suggests it's "phoning home", very likely transmitting personal data. A Google image search on "android malware installation popup" gives you some examples:
browser popups wanting you to install an "update" (click images for larger variants
These candidates are easy to tell: you will never be informed on a system upgrade via browser popup, and there is no such thing like an update for your battery module coming this way (apart from which: the second screenshot proves its "fake-ness" by being a bad translation with grammar mistakes – another indicator to check for). A little protection can be gained from disabling "unknown sources" in your Android settings: that should keep installations from other than "official sources" (Playstore, your vendor) out. So if something claims to be an official update, but on installation you get told to enable "unknown sources" – be sure to cancel, as that's a fake and will almost for sure install malware.
But not all candidates are as easy to tell. Some even manage to place themselves as notifications in your notification area. A few rules on those:
- official updates never come as "browser popups" on visiting some site. They are usually shipped as ota-update, or are to be installed via your device manufacturer's special tool (e.g. Samsung Kies)
- if something tells you it has detected a "virus" – forget it, there is no virus for Android.
- if unsure on its genuity: long-press the notification and check which app caused it. If it's an app that didn't come pre-installed: get rid of that app ASAP, it might do other evil things.
- if you still cannot tell: ask in a forum where your device is dealt with, ask the manufacturer: is there an update deployed currently?
If you're already sitting in the mess (having that fake installed), the safest way out is a factory-reset. You never can be sure where on your device the malware has spread. Hopefully it didn't manage to exploit some security hole and install itself as system app; if it did, the only thing to cure your device is a fresh firmware install.