Given a polarized abelian variety (X, L) over a field k, the automorphism group Autk( X, L) of (X, L) is known to be finite. Hence, it is natural and interesting to ask whether a given finite group can be realized as the automorphism group of a polarized abelian variety over a field k or not. For the case when the ground field k is a finite field, not much was known in this direction, even for the case of X being abelian surface. For this reason, in this dissertation, we give a classification of finite groups that can be realized as the automoprhism group of a polarized abelian surface over finite fields which are maximal in the isogeny class of another abelian surface over the same field. We use various kinds of mathematical results such as Honda-Tate theory and quaternionic matrix representation theory to obtain such a classification.