A question of B. Plotkin about the semigroup of endomorphisms of a free group (Q2781237)

The author proves that if \(F\) is a free group of finite rank \(n\geq 2\), then any automorphism of \(\text{End}(F)\) (the semigroup of endomorphisms of \(F\)) is actually a conjugation by an endomorphism of \(F\). This means that if \(T\colon\text{End}(F)\to\text{End}(F)\) is an automorphism of \(\text{End}(F)\), then there is an \(\alpha\in\Aut(F)\) such that \(T(\beta)=\alpha\circ\beta\circ\alpha^{-1}\) for all \(\beta\in\text{End}(F)\). The proof uses the completeness of \(\Aut(F)\) which was established by the author and \textit{J. L. Dyer} [J. Lond. Math. Soc., II. Ser. 11, 181-190 (1975; Zbl 0313.20021)].