Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford's pi-biproduct. Firstly, we discuss the endomorphism monoid End pi-Hopf(A x H, p) and the automorphism group Aut pi-Hopf(A x H, p) of Radford's pi-biproduct A x H = {A x H alpha}alpha is an element of pi, and prove that the automorphism has a factorization closely related to the factors A and H = {H alpha}alpha is an element of pi. What's more interesting is that a pair of maps (FL, FR) can be used to describe a family of mappings F = {F alpha}alpha is an element of pi. Secondly, we consider the relationship between the automorphism group Aut pi-Hopf(A x H, p) and the automorphism group Aut pi-YD-Hopf(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{Aut}_{\pi\text{-}\mathcal{Y}\mathcal{D}\text{-Hopf}}(A)$$\end{document} of A, and a normal subgroup of the automorphism group Aut pi-Hopf(A x H, p). Finally, we specifically describe the automorphism group of an example.