A couple of answers appeared whilst I was typing mine. Why is this complicated?
///Barsel //What's the difference between taxi taking you for an eye appointment or a friend?//
Hard to tell from the Government guidelines.In essence I cannot see any difference,maybe it could be covered by the 'charitable act' exemption.///
From the guidance (though not the law and it would be hard to see how they would enforce any transgression):
"Car sharing is not permitted with someone from outside your household or your support bubble unless your journey is undertaken for an exempt reason. For example, if car sharing is reasonably necessary as part of your work."
A journey for a medical appointment is an "exempt reason".
//The logical thing to do would be to contact the local police station and ask them for guidance.//
Unless you're in Derbyshire. In fact, I'd stick to my tried and trusted precaution which is that the very last person you should rely on for legal advice is a serving police officer (whether in Derbyshire or not).
And here's the big danger:
//The main reason I'm asking is because I know my friend would find it very difficult if she had to pay a fine.//
Recent events have shown that some people have now become frightened to leave home. Their fear is not of contracting the virus but of being quizzed by the police as to the purpose of their journey and of being "fined" if their response is not deemed acceptable. So Barsel's friend might choose to forego her medical appointment with all the consequences that may entail. That is why I have made such a fuss over the two ladies walking in Derbyshire. This country is descending into a state of apprehension and torpor, not because of the virus, but because of the restrictions they face (or in this case, do not face) and the behaviour of the police.