Automorphisms of metacyclic groups

Abstract: A metacyclic group $H$ can be presented as $\langle \alpha,\beta\mid \alpha^{n}=1, \ \beta^{m}=\alpha^{t}, \ \beta\alpha\beta^{-1}=\alpha^{r}\rangle$ 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.