Actually where it is PAYE, officially the burden does still lie with the taxpayer. The employer does deduct the tax but where there is an error the responsibility for repayment is with the taxpayer - usually I think they adjust the figure and reclaim it over the next tax year however where that can't be done then the employee can be hit with a bill.
Whether they do that in this instance it's not possible to tell but I can't imagine they have gone to all this trouble not to bother.