let $gamma$ be a normal subgroup of the full automorphism group $aut(g)$ of a group $g$, and assume that $inn(g)leq gamma$. an endomorphism $sigma$ of $g$ is said to be {it $gamma$-central} if $sigma$ induces the the identity on the factor group $g/c_g(gamma)$. clearly, if $gamma=inn(g)$, then a $gamma$-central endomorphism is a {it central} endomorphism. in this article the conditi...