She hasn't got many friends, yet everyone likes her.
Yet is a conjunction here to introduce something surprising after what you have just said. I feel it means something like in spite of a fact or at the same time. It sounds better than the others in this sentence because these sentences are about one person and are very related to each other.
In spite of the fact that she hasn't got many friends, everyone likes her.
She hasn't got many friends; however, everyone likes her.
However is an adverb here and should come after a dot or semicolon. Like "yet", it introduces something that seems surprising or very different from what you have said. It doesn't sound well in the sentence above, but it intends to add an unexpected fact.
However is more formal than the others . It is also more suitable to connect two long and more independent sentences, while "but" is used as a conjunction for shorter opposite and joint clauses or even two nouns or two adjectives.
She hasn't got many friends, but everyone likes her.
As I said but is a conjunction and in this sentence acts similar to however, but it's more suitable for these two short sentences. To me, it means "no matter"