On intrinsic Hodge-Tate-ness of Galois representations of dimension two

In the present paper, we first prove that, for an arbitrary reducible Hodge-Tate p-adic representation of dimension two of the absolute Galois group of a p-adic local field and an arbitrary continuous automorphism of the absolute Galois group, the p-adic Galois representation obtained by pulling back the given p-adic Galois representation by the given continuous automorphism is Hodge-Tate. Next, we also prove the existence of an irreducible Hodge-Tate p-adic representation of dimension two of the absolute Galois group of a p-adic local field and a continuous automorphism of the absolute Galois group such that the p-adic Galois representation obtained by pulling back the given p-adic Galois representation by the given continuous automorphism is not Hodge-Tate.