Claiming back double tax on share options

Hello

I have a client who works in the UK but has exercised stock options through the US holding company. The gains have been taxed on her payslip, however she has payed tax in the UK through a PAYE deduction and her payslip also indicates that she has had tax withheld on the share option gains for the US. 

WHat is the best way of recouping this double tax? Is it through the UK tax return or do we apply to the US authorities.

Thanks very much!