Summary: A metacyclic group \(H\) can be presented as \(\langle\alpha,\beta\colon\alpha^n=1\), \(\beta^m=\alpha^t\), \(\beta\alpha\beta^{-1}=\alpha^r\rangle\) for some \(n\), \(m\), \(t\), \(r\). Each endomorphism \(\sigma\) of \(H\) is determined by \(\sigma(\alpha)=\alpha^{x_1}\beta^{y_1}\), \(\sigma(\beta)=\alpha^{x_2}\beta^{y_2}\) for some integers \(x_1\), \(x_2\), \(y_1\), \(y_2\). We give sufficient and necessary conditions on \(x_1\), \(x_2\), \(y_1\), \(y_2\) for \(\sigma\) to be an automorphism.