Let $G$ be a finite perfect group, and let $N$ be the solvable radical of $G$. If $G/N$ is a non-abelian simple group, then is it true that $N$ is contained in the Schur multiplier of $G/N$?

If this is not true in general, then does it hold at least in case $G/N$ is either of type ${\rm PSL}(2,2^p)$ ($p$ prime) or isomorphic to one of ${\rm PSL}(2,7)$ or ${\rm Sz}(8)$?

Furthermore, can the finite perfect groups whose quotient modulo their solvable radical is isomorphic to one of the simple groups mentioned in the previous paragraph reasonably be classified?