Kourovka Notebook Problem 21.134

For a finite group $G$, let the type of $G$ be the function on positive integers whose value at $n$ is the number of solutions of the equation $x^n=1$ in $G$.

a) Is it true that a group having the same type as a group with trivial solvable radical must also have trivial solvable radical?

b) Is it true that a group having the same type as an almost simple group must be isomorphic to it?