On central endomorphisms of a group

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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 conditions under which a $Gamma$-central endomorphism of a group is an automorphism are investigated‎.