Does UK rail station's ticket machine sell group tickets?

OK, the official answer from the staff at Cambridge station is "yes, you can buy a GroupSave ticket from the machines". I tried buying a GroupSave (GS) return to Ely. It isn't straightforward.

Firstly, simply telling it you have three people isn't enough, you have to press "Railcards" to access the GroupSave option. Secondly, although the machine says the ticket is valid on all services, according to the staff it is lying, and GS tickets CBG-ELY are not valid on Cross Country services (ie, those that are going to/from Birmingham). Similar issues arise with many other GS tickets.

As for hours, the ticket machines are accessible "all the hours the station is open", which is from shortly before the first train to shortly after the last one. According to my helpful staff member, this is roughly 0400-0115, which is a wider set of hours than those for which the ticket office is open

As another caveat, if you try to buy tickets after midnight and before the station shuts, the machines will try to sell you a ticket for the day that has just passed (because tickets for day X are valid until the small hours of day X+1); you have to buy a ticket for "tomorrow" to get one valid for the actual day in which you find yourself. But by 0400 the machines have often caught on, so if you then ask for a "tomorrow" ticket out of force of habit, you will get one for the following day.

In short, the advice of the (super-friendly and helpful) staff member is "don't buy GroupSave tickets from the machines, come to us". But if you really want to, then that is how and when you can do it. You might also want to try buying them in advance via National Rail Enquiries, and just collect them from the machine; apparently NRE does know which trains GS tickets are valid on.