Great question. Just to add to the already excellent answers; there is an SAS abbreviation which has always served me very well, and it applies here. PLAN your course of action;
- Protection: The safety of the group, and the individuals within the group, is of greatest importance. Ensuring you protect the group and it's members is the first priority. It depends on the location, and the weather for you to decide the best course(s) of action to ensure that.
- Location: Exposure is a merciless enemy. High winds, temperature extremes and terrain must all be considered. If you are in a particularly hostile environment, it may be best to set up camp, as above, to ensure protection of the ill party and the group. Alternatively, given fair weather and relatively accessible terrain, pressing on to the nearest safe point may be best.
- Acquisition: Getting the stuff you need; food, water, (in this case, medicine) is next. Once the P and L have been decided upon, it's time to acquire the necessaries for treatment, even if that is paracetamol or ibuprofen. A warm tent, plenty of fluids, in-depth assessment of the ill party all fall under this category. Don't forget you may have one ill person in your immediate concerns, but the group will soon grow hungry, thirsty, and potentially tired while waiting for treatment of the individual.
- Navigation: By this point, you (or a senior) has taken definite action to protect all those involved, you have decided on your location (stay, go on, pitch camp, etc) and you have acquired the necessary materials, or at least decided how to do so. Now you must decide on navigation. If you are pressing on, where are you going? If you're staying put, or pitching camp, when will you move on, who will it be? The whole party or an individual? Do you know where you are going - very important to establish this at the outset and make it clear to all involved. If you don't know, be honest.
PLAN applies to group-size decisions, or even your own actions within the group. It's an excellent concept to remember because it forces you to stop and think about a course of action, and then act on, and ideally stick to the plan.
Whatever you decide upon, make it clear to all parties involved. A deteriorating patient being dragged down a mountain will inevitably cause distress (and potentially, anger) amongst the group; very dangerous in a hostile environment.
In my observation, men will usually want to press on in the event of illness, or injury, if they are able to. Contrasted with women, who are more likely to opt for a rest or to camp up. You or a senior member might have to take the decision on behalf of an ill/injured male companion, for his, and the group's safety.
Pressing on in the event of a persistent high fever is almost always a bad idea. The heart works harder when the body temperature increases, and continuing, even at a reduced pace, increases the work it has to do even more. That has knock-on effects, and may even cause damage to the patient's heart. Knock-on effects include nausea, exhaustion, vomiting, dehydration, etc, etc.
In almost all cases, hydration is key - provided the water is clean. Do not introduce fluids from an non-trustworthy source.
And lastly, throw in a pinch of kindness. The dude is ill, how would you feel if you were pressed, or expected to continue under similar conditions. Comfort, and make them comfortable, if you can. Reassurance and a good chat will also do wonders in most cases. Also don't underestimate the heartening effect of a good brew for all parties involved. By "brew", I mean tea or similar, NOT alcohol!
Hope for the best, PLAN for the worst. :)