A group $G$ is locally solvable if all finitely generated subgroups are solvable.
A group $G$ is locally finite if all finitely generated subgroups are finite.
A group $G$ is virtually locally solvable if it has a locally solvable subgroup of the finite index.
Let be $R(S)=\left\langle T\,;\,T\trianglelefteq G\,,\,T\text{ locally solvable }\right\rangle $
My question are:
1)Is $\,R(S)\,$ locally solvable?
2) If 1) is true: $G$ locally finite, $R(S)$ locally solvable and $G/R(S)$ virtually locally solvable $\Rightarrow G$ virtually locally solvable?