Perfect groups whose character degrees square divide its order This post is an extension of that one from the non-abelian finite simple groups to the finite perfect groups. According to the mentioned post and its second comment, for all non-abelian finite simple group $G$, there is an irreducible complex character $\chi$ such that $\chi(1)^2$ does not divide $|G|$. The proof uses the classification of finite simple groups (CFSG), but we are interested in a CFSG-free proof, so if you find one, please post an answer to the mentioned post. Anyway, here we are interested in the finite perfect groups:
Question: Is there a finite perfect group $G$ such that for all irreducible complex character $χ$ then $\chi(1)^2$ divides |G|?
Bonus question 1: Is there such a group which is a direct product of non-abelian finite simple groups?
Bonus question 2: If so, what is the minimum number of components for such a direct product? Two?
Above bonus questions can be seen as a game with the prime factorization of the order of the non-abelian finite simple groups and their character degrees. Let us call a finite group $G$ thin if for all prime $p$ then there is an irreducible complex character $\chi$ such that $\nu_p(\chi(1)) = \nu_p(|G|)$, where $\nu_p$ is the p-adic valuation. The seven first non-abelian finite simple groups ($\mathrm{PSL}(2,q)$ with $q=5,7,9,8,11,13,17$) are thin. We wonder whether every $\mathrm{PSL}(2,q)$ is thin. Anyway, for our purpose here, we must consider non-thin groups only. A finite group $G$ will be called $p$-fat if for all irreducible complex character $\chi$ then $\nu_p(\chi(1)) < \nu_p(|G|)$. Note that a finite group is non-thin if and only if it is $p$-fat for some prime $p$. The first non-thin non-abelian finite simple group is $A_7$, which is $2$-fat. Now the notion of $p$-fat group is not enough for our need. A finite group $G$ will be called $p$-superfat if for all irreducible complex character $\chi$ then $2\nu_p(\chi(1)) < \nu_p(|G|)$. The group $A_7$ is $2$-superfat. The next $p$-superfat non-abelian finite simple group is $M_{22}$, which is also $2$-superfat. We wonder whether for all prime $p$ there is a $p$-superfat non-abelian finite simple group. Anyway, the goal here is to find relevant direct product of such superfat groups.
 A: I cannot give a complete answer, but here are some thoughts.
In "Gagola, Stephen M., Jr.; A character theoretic condition for $F(G)>1$. Comm. Algebra 33 (2005), no. 5, 1369-1382." the following result is proven (relying on CFSG):

Theorem: Suppose that $G$ is a nontrivial finite group such that $\chi(1)^2 \mid |G|$ for all complex irreducible characters $\chi$ of $G$. Then $G$ has a nontrivial abelian normal subgroup.

So the answer to your bonus question is no.
Now for an example as in your main question, the group $G$ must have a nontrivial abelian normal subgroup $N$. If you can prove that the "squares of character degrees divide order" property holds for $G/N$ as well, it would follow by induction that no such $G$ exists.
Another related result is in "Gagola, Stephen M., Jr.; Lewis, Mark L. A character-theoretic condition characterizing nilpotent groups. Comm. Algebra 27 (1999), no. 3, 1053-1056." (This also relies on CFSG.)

Theorem: Suppose that $G$ is a finite group. Then $G$ is nilpotent if and only if $\chi(1)^2 \mid [G:\ker \chi]$ for all complex irreducible characters $\chi$ of $G$.

