Why is fuel cost the only running cost taken into account, OP?
As per latest available AA figures, fuel is about 60% of the totsl running costs, per mile. If fuel is 10p per mile, total running costs is about 18p. These costs fo not include buying/financig the car, insurance, etc and other standing costs incurred whether the car is used or not.
So, at those rates, 50km is about 31 miles, so at 18p, cost for that 50km per day is £5.60 approx.
If I were receiving the lift, I'd regard that as good value since it's still a saving over getting the bus, and is far more convenient to me that using public transport. Frankly, I'd pay a premium over the cost of the bus for the convenience of the car ride.
If I were the driver, I'd regard the passenger paying the cost of running for the miles he gets as "fair". I am, after all, providing him a service and saving him money. But personally, I would care if I got paid or not as I'd be doing the journey anyway. I wouldn't make an issue of costs st all.
However, once it had come ip, if I got the impression that the passenger was being tight about it, it would likely be the end of my willingness to act as a taxi whatever he paid. Being paid, even getting a contribution to costs wouldn't be on my mind, but having to negotiate a figure would depress me.
If nothing was offered, okay. If I was expecting a contribution, I'd have said so. If a decent conttibution was offered, okay. If a low-ball offer was made, I'd be offended and find a reason/excuse to stop offering the lift at all.