That would be my understanding as you will be dealing with the car purely as a business transaction by "recharging the lease payments", perhaps in the form of a hire and there would be no use of the car, private or otherwise from the original lessee company. So they (the original lessee) would claim 20% as input vat and charge 20% in turn to the sister company.
If the sister company has private use for the car, they would claim 50% of the input vat. If there is no private use all the vat may be reclaimable see https//www.gov.uk/reclaim-vat/cars
Hope this helps