On the universal <inline-formula><tex-math id="M1">$ \alpha $</tex-math></inline-formula>-central extensions of the semi-direct product of Hom-preLie algebras

We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal \begin{document}$ \alpha $\end{document} -central extensions of \begin{document}$ \alpha $\end{document} -perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an \begin{document}$ \alpha $\end{document} -cover with certain constraints. We provide some necessary and sufficient conditions about the universal \begin{document}$ \alpha $\end{document} -central extension of the semi-direct product of two \begin{document}$ \alpha $\end{document} -perfect Hom-preLie algebras.