Infinitely powered endomorphism of a group

Definition

Suppose $G$ is a group and $f$ is an endomorphism of $G$. We say that $f$ is infinitely powered if its [[[powering threshold for an endomorphism of a group|powering threshold]] is $\mathrm{\infty }$. Explicitly, this means that for all natural numbers $n$, the image $f^n(G)$ is powered for all primes less than or equal to $n$.