A weighted composition operator is an operator generalizing the notion of multiplication and composition operators. The theory of weighted composition operators on spaces of holomorphic functions has recently been expanding into a very active field of mathematics. In particular, we will be interested in the class of weighted Hardy spaces of holomorphic functions, as formally introduced and explored in a seminal paper of Shields from 1974. In this thesis, we investigate some particular properties of weighted composition operators acting on the class of weighted Hardy spaces on the open unit disk. The main goal is to determine the behaviors of the symbols of co-isometric, unitary, invertible, and isometric weighted composition operators acting on spaces of our interest. We modify the result by Martin, Mas, and Vukotić from 2020, to present the conditions on the symbols of co-isometric and unitary weighted composition operators on the class of weighted Hardy spaces on the open unit disk. We then explore the result by Arévalo and Vukotić from 2020, to see the characteristics of the symbols of invertible weighted composition operators on the class of weighted Hardy spaces on the open unit disk. In addition, we will consider the other direction of the above result on a certain subclass of such spaces, namely on the disk automorphism invariant subspaces. Although Isometric weighted composition operators are slightly more difficult to analyze compared to the previous properties. Nevertheless, we were able to adapt the result by Jaoua from 2010 to our context, and describe the conditions on the symbols of isometric weighted composition operators on a subclass of the class of weighted Hardy spaces on the open unit disk. Furthermore, we also investigate isometric weighted composition operators defined by monomials on the class of weighted Hardy spaces on the open unit disk.