Whisper doesn't, but WhisperX <https://github.com/m-bain/whisperX/> does. I am using it right now and it's perfectly serviceable.

For reference, I'm transcribing research-related podcasts, meaning speech doesn't overlap a lot, which would be a problem for WhisperX from what I understand. There's also a lot of accents, which are straining on Whisper (though it's also doing well), but surely help WhisperX. It did have issues with figuring out the number of speakers on it's own, but that wasn't a problem for my use case.