An Aufnahmeprüfung for primary school? Is that really a thing, in Basel? I thought that, all across Switzerland, children were simply automatically allocated a place in primary school.
I assume you're talking about going up to Secondär. This applies to BS and BL (and indeed all German speaking cantons).
Many parents just get a tutor in when the kids are 10 or 11. The evaluation begins at the start of year 6. If your child is already in that, you're too late to change anything. But they can still switch levels even in secondary school. My youngest was in niveau A, and got to gym.
However, we never bothered with extra tutoring or additional classes.
Ah, an Aufnahmeprüfung to get into gymnasium after primary school. I see now. Yes, that's common. I had thought OP had meant an exam to get into primary school itself.
There is no Aufnahmeprüfung in Basel or Baselland. The Übertritt in BS (Sek A, Sek E, Sek P) is based on the grades of children. In BL, the "Klassenlehrperson" places the child in Sek A, E or P (still based on grades, but with more leeway, IIRC). There is, however, an optional "Übertrittsprüfung" in both cantons where you can send your child to if you're not happy with the decision. The Übertritt into Gymnasium only happens after Sekundarschule, again based on grades (with some special rules, at least in BL). No "Übertrittsprüfung" there. Either you make it or you don't.
Funny though. My daughter is training as a Primary teacher, and she recognised the term immediately (and supplied me with my earlier reply), so it seems to be still in use, even though the exam has been abolished. All the German speaking cantons have agreed to harmonise their education systems and so have essentially the same rules.
In BS access to gymnasium and to the different streams or Sek is regulated by the marks in both semesters (there is an equation to calculate it). If you disagree with the results you can do the Aufnahmeprufung. Chances to change the marks verdict are very low indeed. That’s what the OP meant I guess
It’s the same in Bern and Solothurn. I teach grade 6 in Kanton Berne. We send maybe one or two kids out of four classes to the exam each year, because the parents don’t accept our decision. Very seldom does the exam actually change the outcome.
My daughter dropped out of gym to do an apprenticeship. Did a BM, and now is studying for a Bachelors. There are many educational routes. Some take longer than others.