"Group" falls under the category of collective nouns. In American English we treat these as singular. In British English they're often treated as plural nouns. See this page for a more detailed explanation: http://www.onestopenglish.com/grammar/grammar-reference/american-english-vs-british-english/differences-in-american-and-british-english-grammar-article/152820.article

come because the subject of the verb is "a group" singular.

It's true that many group nouns can be plural in British English, but in this case whether 'group' is singular or plural completely irrelevent. It's not a case of using one type of English or determining whether a noun is singular or not, but using the subjunctive mood. http://www.englishpage.com/minitutorials/subjunctive.html In the subjunctive, the verb is always unconjugated.

Just to add something now that I've seen your example: "I suggest that your group stay with the families of the students from our school." This is actually a subjunctive mood, so there is no way of telling whether the group is being treated as a singular or plural noun. The verb would be 'stay' (the base form) regardless of the subject.

It is, and as such it is treated as grammatically singular in US.E, but as either singular or plural in GB.E. If you are using British English, the choice of singular or plural depends on what seems most logical in the context, and whether you are considering the group as one entity or as a collection of individuals.