Recall R. Thompson's group $F$ acting on the interval $[0,1]$: it consists of piecewise linear oriented maps with slopes a power of $2$ and dyadic breakpoints.
Is every finite subquotient (= quotient of a subgroup) soluble?
[my money is on "yes". Finite quotients of $F$ itself are abelian. I know subgroups of $F$ that are made of copies of $F$ and of soluble groups, but nothing more exotic.]