It is known that the automorphism group of any projective K3 surface is finitely generated [24]. In this paper, we consider a certain kind of K3 surfaces with Picard number 3 whose automorphism groups are isomorphic to congruence subgroups of the modular group $PSL_2(\mathbb{Z})$. In particular, we show that a free group of arbitrarily large rank appears as the automorphism group of such a K3 surface.