On automorphisms and endomorphisms of $CC$ groups

We consider the question of describing the automorphisms of semigroups $\mathrm{End}(G)$ of groups $G$ having only cyclic centralizers $(CC)$ of nontrivial elements. In particular, we prove that each automorphism of the automorphism group $\mathrm{Aut}(G)$ of groups $G$ from this class is uniquely determined by its action on the elements from the subgroup of inner automorphisms $\mathrm{Inn}(G)$. Note that, for instance, absolutely free groups, free periodic groups of large enough odd periods, $n$-periodic and free products of $CC$ groups also are $CC$ groups.