Are the perfect groups linearly primitive? A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial.
A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation.  
Question: Are the perfect finite groups linearly primitive?
Remark: the finite simple groups are perfect and linearly primitive.

No. Any linearly primitive group must have cyclic center by Schur's lemma, but there's no reason a perfect group should have this property. I think that, for example, the universal central extension of $A_5 \times A_5$ is perfect but has center $C_2 \times C_2$. 
