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Let Γ be a normal subgroup of the full automorphism group Aut(G) of a group G ‎, ‎and assume that Inn(G)≤Γ ‎. ‎An endomorphism σ of G is said to be {\it Γ -central} if σ induces the the identity on the factor group G/C G (Γ) ‎. ‎Clearly‎, ‎if Γ=Inn(G) ‎, ‎then a Γ -central endomorphism is a {\it central} endomorphism‎. ‎In this article the conditions under which a Γ -central endomorphism of a group is an automorphism are investigated‎.