There may be exactly $n$ $Q$-points

We generalize the main result of arXiv:2505.17960 and show the consistency of the statement ``There are exactly $n$ $Q$-points up to isomorphism"for any finite $n$. Furthermore, we show that the above statement for $n=2$ can alternatively be obtained by a length-$\omega_2$ countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.