An extension of a theorem of Neumann

In the present article, we prove that every countable infinite group $G$ is embeddable into a countable infinite simple group $\overline{G}$ such that every equation of the form
$$w(x_1,\dots,x_n)=g,$$
is solvable in $\overline{G}$, where $w$ is a nontrivial reduced group word in variables $x_1,\dots,x_n$, and $g\in G$.