On the weak $\pi$-potency of some groups and free products

Let $\pi $ be a set of primes. A group $G$ is weakly $\pi$-potent if $G$ is residually finite and, for each element $x$ of infinite order in $G$, there is a positive integer $m$ such that, for every positive $\pi$-integer $n$, there exists a homomorphism of $G$ onto a finite group which sends $x$ to an element of order $mn$. We obtain a few results about weak $\pi$-potency for some groups and generalized free products.