The following well known theorem can be found in many books on character theory:
Let $\chi$ be a faithful character of a finite group $G$ and suppose that $\chi(g)$ takes on exactly $m$ different values for $g \in G$. Then every $\psi \in \operatorname{Irr}(G)$ is a constituent of one the characters $\chi^j$, for $0 ≤ j < m$.
This particular theorem gives no information about the multiplicities of the constituents of the $\chi^j$.
My question is, are there any results which give more detailed information (dimension, etc... ) about the subspace of the space of class functions spanned by powers of a faithful irreducible character $\chi$, perhaps given some conditions on $G$? I'm interested in the question in general, but especially in the case where $G$ is a $p$-group.
Edit:
As an example of what I mean, the group $S_4$ has 5 conjugacy classes, and has a faithful irreducible character $\chi$ which takes on the values $[3,-1,-1,0,1]$. Fixing an ordering for the conjugacy classes of $S_4$, I'm thinking of this as a vector in $\mathbb C^5$. The powers of $\chi$ are of the form $\chi^j = [3^j,(-1)^j,(-1)^j,0,1]$. Now $1,\chi,\chi^2,\chi^3$ span a 4 dimensional subspace of $\mathbb C^5$, and if my calculations are right, all $\chi^j$, $j >3$ can be expressed as linear combinations of lower powers of $\chi$.
Another example I tried was an extraspecial group of order 27. Here I found 11 conjugacy classes, and a faithful irreducible character $\chi$ of degree 3 whose powers only spanned a subspace of dimension 4.