How your friend can possibly get what she wants
E14 positions are very rare in German academia. You hardly ever see them advertised (I would very roughly estimate you get one E14 for thousand E13s).
Instead they are almost always acquired through promotions in later career stages, which are also rare.
Moreover, goodwill alone doesn’t suffice as the university must have the funds to back this up.
It already surprises me that the HR department did not directly send the group leader somewhere else to secure funding first.
Threats only help if following up on them would harm the person who can give you what you want, and in your friend’s case this may not be the group leader.
If your friend really wants to pursue E14, I suggest that she finds other people who got such a promotion. This would usually be staff scientists, habilitated people (“Privatdozent”), etc. Also ask the department or faculty whether it has specific funds for this, because without them this is likely hopeless.
Finally, as already suggested by other answers, I would start with smaller levers than quitting such as refusing extra responsibilities.
… but does she actually want this?
Most importantly, your friend should think strongly about whether a salary increase is actually what she wants, for several reasons:
The academic apparatus is not geared to rewarding people with money. In most fields, people who care about money leave academia, and academia doesn’t need people who are in it for the money. (This is not supposed to say your friend is obsessed with money or similar.) The rewards of good work in academia are rather that you get to stay in academia, get more funds (not for salary), etc., in particular before the professor stage. I would therefore suggest that your friend considers seeking alternative “rewards” such as a permanent position, personnel to help her advance her own science, etc. (Paradoxically, funds for this may be more easily available.) This is particular important as:
The main risk I see for your friend is this:
If she wants to pursue a classical academic career, one of the main thing she needs to show is her own research output.
Taking over the group leader’s duties can give relevant experience in teaching, supervising, etc. and those have value in future applications, but they can only compensate so much for a lack of research.
(Mind that this is a vague guess:
Your friend may actually get a lot of career-relevant research under her name, in which case she may be already in for a long-term reward, as she has well rounded group-leader experience.)
A salary increase does not address this and is worth very little if your friend has to leave academia – probably around the time when the benefits of the salary increase would kick in, because:
The actual short-term increases to salary may be rather low because the German public salary system has two axes, one for responsibility/qualification and one for job experience. If you friend progresses on the first axis, she will probably take a step back on the other axis. She cannot lose salary (by law), but the real benefits may only manifest as she steps up the experience ladder faster and with more benefits. Details depend on her current experience. This contributes to why this is usually done only in latter career stages (for postdocs and similar).
Possible strategies
If it did not happen already, I suggest that your friend has a talk with her group leader about how the extra duties affect her career chances and what benefits she can expect in compensation (other than extra salary). She should also talk to other professors in the field about this. The field is important here since different fields differently weigh publications, supervision experience, teaching experience, etc.
In German academia, successfully applying somewhere else may result in a promotion (or permanent position) at your current employer and is a standard way to get this – usually but not only at the professor level.
Instead of threatening to quit, I therefore suggest that your friend takes a look at the relevant job market and applies for suitable jobs, in particularly permanent ones.
An external job offer or even being invited to interviews are a much more palpable threat than just saying you intend to quit.
Moreover, this lowers the risk for your friend to end up without a job or having to concede that her threat was empty.
Your friend has to be the judge whether to seek her group leader’s support for this.
They may support this, in particularly if leaving the group is not the actual goal – but they may also equate it to a threat to quit or feel betrayed.