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 $\Gamma$-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 central endomorphism. In this article the conditions under which a $\Gamma$-central endomorphism of a group is an automorphism are investigated.
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