This post is a sequel of this one.
Question: Is there a finite perfect group $G$ such that for all (non-trivial) irreducible complex character $\chi$ then $\chi(1)^2$ divides $|G|$ and $\chi(1)$ is not a prime-power?
The post mentioned above got an answer containing a construction starting from the simple group $A_5$ whose character degrees are $[[1,1],[3,2],[4,1],[5,1]]$, and finishing with a perfect group $G$ with character degrees $$ [ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 7 ], [ 10, 4 ], [ 12, 10 ], [ 15, 14 ], [ 20, 59 ], [ 30, 164 ], [ 60, 2651 ] ]$$ Observe that the character degrees of $G$ are multiples of the ones of $A_5$. If this happens also by starting from an other simple group, then I suggest to try with $A_7$ whose character degress are $[ [ 1, 1 ], [ 6, 1 ], [ 10, 2 ], [ 14, 2 ], [ 15, 1 ], [ 21, 1 ], [ 35, 1 ] ]$, i.e. no prime-power. But it is not clear to me how to make above construction from $A_7$ in practice (assuming that my expectation is true).
A positive answer to this question will answer a problem in the theory of fusion rings.