Is "There was a group of people who were enjoying it" grammatically correct?

If you want enjoying to apply to the group as a whole, you have to use that rather than who.

There was a group of people that was enjoying it.

When you use who, it refers to people, which is plural, so it needs a plural verb. Switching to that changes the reference to group, which is singular.

You can also say:

There was a group of people that were enjoying it.

There is a very subtle difference, I believe. The first emphasizes that the group as a whole is enjoying it. The second emphasizes that a number of people are enjoying it, and then defines the group in terms of those people.

Whenever I get stuck into such a situation, I think of my mother tongue (Urdu_Pakistan). It gives me two senses, both of them seem true and do not sound ambiguous to me. I think whether it is 'was' or 'were', it depends upon you what you want to get out of it.

Question: What is correct in the following example? Who or that?

one group who will conduct the experiment or one group that will conduct the experiment

..............There...............was... {a group of people who were enjoying it.}

Adverb(complement)... verb......................... subject

Thus we have

There was a group {of [people who were enjoying it]}.

[people who were enjoying it] is a noun phrase made up of a noun (people) + relative clause (who were enjoying it).

{of [people who were enjoying it]} is a prepositional adjectival clause modifying "group"

Thus because the referent of "who" is "people", "were" is correct.