Crossed product Hom-Hopf algebras and lazy 2-cocycle

Let $(H,\a)$ be a Hom-Hopf algebra and $(A,\b)$ be a Hom-algebra. In this paper we will construct the Hom-crossed product $(A\#_σH,\bø\a)$, and prove that the extension $A\subseteq A\#_σH$ is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make $(A\#_σH,\bø\a)$ a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on $(H,\a)$.