Permutations involving squares in finite fields

Let $ p $ be an odd prime and let $ \mathbb{F}_p $ be the finite field of $ p $ elements. In 2019, Sun studied some permutations involving squares in $ \mathbb{F}_p $. In this paper, by the theory of local fields we generalize this topic to $ \mathbb{F}_{p^2} $, which gives a partial answer to the question posed by Sun.