Academic with fellowship cannot land a permanent position in her department. Should she stay or move elsewhere?
Universities in the West are generally quite reluctant to hire anyone into permanent positions who had been graduate students or postdocs or otherwise temporary employees at that same university. This is fundamentally based on the idea that we don't want "inbreeding" of ideas and that it benefits departments and universities to get in people with other perspectives. As a consequence, there may not be merit-based reasons why she isn't hired -- it may all be due to not wanting to hire your own disciples.

The solution is of course to go elsewhere -- as almost all of us had to do at some point. Everyone understands that to be the system, and students need to know it as well. In many cases, one can make practical arrangements where students come along, or where someone who leaves can still advise students at their old university until they are graduated. (I have done both.)