Both answers are mainly right, and yet the second pair of sentences you use to highlight your question embodies a problem.
In your second example, “the group is helping one another to become better students” strikes a discordant note in my ear. This discord would be resolved if you said: “the **members of the ** group are helping one another ...”. But this phrasing is a little laboured. If the group had a name, the problem also disappears: “Alpha Kappa are helping one another ...” is fine. A named group of people feels like the plurality of its members in a way that the word ‘group’ does not.
The problem is the use of a collective noun group with a reciprocal in the same clause.
oxforddictionnaries.com provides an excellent discussion of whether we should use singular or plural verbs with collective nouns, supported by its vast database of usage. It points out that usage is changing, and that British and American English differ. I noticed this every time Google’s US spelling checker kept trying to change “staff” to “staffs” in sentences like “staff know their pupils well.” (Before I retired I was a Her Majesty’s Inspector of schools and cursed the spell checker quite a bit!). Americans use a singular with collective nouns rather more than Brits do.
So your second pair of sentences both ring grammatical alarm bells for me because two potential singular/plural dissonances are combined in the same sentence.
If only there were another word for students at the end of the sentence, we could start the sentence with “The students in my class”, and the problem would vanish. Well, could we use better learners? That might do. Or possibly we could replace “to be better students” with “to study more effectively”. That would quieten my alarm bells.
Why do I go to this length? Well, I am illustrating a general principal, which is to avoid getting yourself into a position where you have a collective noun and a reciprocal in the same sentence. Find other ways of saying the same thing if you possibly can!
Why? Because grammar is an evolutionary phenomenon, just like all other aspects of life, and, as it evolves, new features appear and snags appear too. Avoid them if you can. Otherwise you are bound to worry someone.
On the other hand, if we all behave in this way, we may be interfering with the natural evolution of language. Hm. There may be a useful question in there somewhere.