Let $G$ be a finite group.
It is well-known that if $G$ is an $M$-group, then $G$ is solvable, due to Taketa. The converse is not necessarily true: $SL(2,3)$ is solvable, but not an $M$-group.However, Dade proved that every solvable group embeds into an $M$-group.
The proof uses induction and the wreath product to show that for any solvable group $G$ there exists an $M$-group $G'$ such that $G$ is a subgroup of $G'$.
I think that results about the structure of $G'$ are probably too difficult, so I am curious about the index of $G$ in $G'$. The use of the wreath product in the existence proof suggests that $G'$ could be very large. Is someone aware of results in this direction? Or perhaps a more constructive proof than Dade's?