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Not sure who you think would pay for it, if not the applicant. If he needs to extend his visa, he needs to apply in the normal way.
..Some students are unable to complete the school placement which is a considerable part of the course, the Dfe are aware that some students will need course extensions and are funding these until the end of December 2021"
The answer from LSBU is a little vague and seems to be about an extension of the course, with DfE covering the tuition fees. I would double check with them what this means for those students who also need a Student visa extension.