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 π-biproduct. Firstly, we discuss the endomorphism monoid End_π-Hopf(A × H, p) and the automorphism group Aut_π-Hopf(A × H, p) of Radford’s π-biproduct A × H = {A × H_α}_α∈π, and prove that the automorphism has a factorization closely related to the factors A and H = {H_α}_α∈π. What’s more interesting is that a pair of maps (F_L, F_R) can be used to describe a family of mappings F = {F_α}_α∈π. Secondly, we consider the relationship between the automorphism group Aut_π-Hopf(A × H, p) and the automorphism group \(\text{Aut}_{\pi\text{-}\mathcal{Y}\mathcal{D}\text{-Hopf}}(A)\) $${\text{Aut}}_{\pi \text{-}\mathcal{Y}\mathcal{D}\text{-Hopf}}(A)$$ of A, and a normal subgroup of the automorphism group Aut_π-Hopf(A × H, p). Finally, we specifically describe the automorphism group of an example.