You didn't ask about it, but there is also a question of whether Friend(s) should be in the genitive (possessive). So in fact there are four possibilities:
singular plural
non-gentive Friend Group Friends Group
genitive Friend's Group Friends' Group
It seems to me that the only one that's absolutely unacceptable in your context is the singular possessive, Friend's Group, because its meaning is really that this is a group associated with one (possibly particular) friend.
Beyond that, in principle there is nothing 'incorrect' about the rest of them. All we can do is, on the one hand, look a the patterns of usage, and on the other hand, recommendations of various authorities.
Patterns of usage
A cursory search would suggest that the genitive is very rarely used for Friend(s) Group. My guess is that this is because Friend(s) Group is used primarily in contexts which are both informal and online, and both of those disfavor the use of the apostrophe (even when it is actually needed). And in the case of Friend(s) Group, since the genitive is not absolutely required, most people seem happy to ignore its possibility.
As far as whether the singular or the plural should be used, there does not seem to be a strong tendency in favor of either.
Recommendations of various authorities
Here I will look at two well-known style manuals. They don't explicitly address the construction Friend(s) Group, but they do address the general question of 'terms denoting group ownership or participation'.
Different style manuals may give different recommendations. The best you can do is pick one and stick with it. I will discuss two quite prominent manuals, Chicago and AP.
Chicago Manual of Style
In this case, we are dealing with the category of 'terms denoting group ownership or participation' (in your case mostly participation), for which Chicago Manual of Style (CMOS) would actually recommend a plural possessive:
Friends' Group
If not plural possessive, it seems CMOS would go with the singular non-possessive:
Friend group
(see taxpayers’ associations/taxpayer associations and consumers’ group/consumer group, below).
Here is the relevant section of CMOS:
(begin quote)
Spelling, Distinctive Treatment of Words, and Compounds
7.27: Possessive versus attributive forms for groups
Although terms denoting group ownership or participation sometimes appear without an apostrophe (i.e., as an attributive rather than a possessive noun), Chicago dispenses with the apostrophe only in proper names (often corporate names) that do not officially include one. In a few established cases, a singular noun can be used attributively; if in doubt, choose the plural possessive. (Irregular plurals such as children and women must always be in the possessive.)
children’s rights (or child rights)
farmers’ market
women’s soccer team
boys’ clubs
veterans’ organizations
players’ unions
taxpayers’ associations (or taxpayer associations)
consumers’ group (or consumer group)
but
Publishers Weekly
Diners Club
Department of Veterans Affairs
In some cases, the distinction between attributive and possessive is subtle. Of the following two examples, only the first connotes actual possession.
the Lakers’ game plan (the team’s game plan)
but
the Lakers game (the game featuring the team)
When in doubt, opt for the possessive.
(end quote)
Associated Press Stylebook
Here are the relevant subsections, from the section on possessives:
DESCRIPTIVE PHRASES: Do not add an apostrophe to a word ending in s when it is used primarily in a descriptive sense: citizens band radio, a Cincinnati Reds infielder, a teachers college, a Teamsters request, a writers guide.
Memory Aid: The apostrophe usually is not used if for or by rather than of would be appropriate in the longer form: a radio band for citizens, a college for teachers, a guide for writers, a request by the Teamsters.
An ’s is required, however, when a term involves a plural word that does not end in s: a children’s hospital, a people’s republic, the Young Men’s Christian Association.
DESCRIPTIVE NAMES: Some governmental, corporate and institutional organizations with a descriptive word in their names use an apostrophe; some do not. Follow the user’s practice: Actors’ Equity, Diners Club, the Ladies’ Home Journal, the National Governors’ Association. See separate entries for these and similar names frequently in the news.
Thus although the AP Stylebook is of no help in choosing between the singular Friend and the plural Friends, whatever you choose, AP Stylebook would recommend against turning it into a possessive. As in many things, so in this the two style manuals give opposite recommendations.
Conclusion
It is not incorrect to use any of the following three possibilities: 1. Friend Group, 2. Friends Group, and 3. Friends' Group. Popular usage disfavors the genitive, 3., even though CMOS would recommend exactly that one. If you decide not to use the genitive, pick either the singular or the plural; there are no deep reasons to favor one of them above the other.