Confused over providing a company car

We are a small start up company hiring a new salesman who needs a vehicle. We are considering the options, but likely  the comapny will lease or hire a vehicle for him to use mostly for work, but some private use.  I'm struggling to understand how the process works.

The employee pays for petrol some of which is for private use. As I understand if the company repays everything he would be hit for an addiitonal P11D charge for the provission of private fuel use.  Since most of his miles will be business I expect this won't be tax efficient. So if we don't pay all of the petrol cost I understand we can repay at up to the advisory fuel rate of say 15p/mile on whatever is documented as business use.  I understand this is free of any tax implication for the employee.

If the above is correct what happens to VAT?  The employee has paid VAT on the fuel purchase so can we claim back VAT on the 15p we pay to him i.e. 2.5p?

I'm assumnig the Fuel Scale Charges I have come across only apply where the company pays all of the fuel costs and therefore where the employee is taking the P11D hit for fuel provision i.e. this is not relevant in this case.

Thanks for any clarification.

Mike