Solvability and solvable radicals from arithmetic invariants

Determine, for inv equal to cs or cd, whether equality of the multiplicity-sensitive invariant inv^*(G)=inv^*(H) implies that H is solvable whenever G is solvable, and whether H has trivial solvable radical whenever G has trivial solvable radical.

Background

The paper gives examples of groups with identical multiplicity-sensitive element-order data but different solvability properties and different solvable radicals. It asks whether analogous phenomena occur for multiplicity-sensitive conjugacy class sizes or character degrees.

The two subquestions test whether these arithmetic invariants preserve solvability and semisimplicity-type information.