In this paper, we study plane quintic curves whose automorphism groups have order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The latter two cases are represented as one-parameter families, where their superspeciality can be explicitly described in terms of a truncation of certain Gaussian hypergeometric series. Applying this characterization, we determine the exact number of isomorphism classes of superspecial plane quintic curves with automorphism groups $\cong \mathbb{Z}/10\mathbb{Z}$. We also provide an efficient algorithm to enumerate such curves with automorphism groups $\cong \mathbb{Z}/8\mathbb{Z}$, and provide the computational results for the range $13 < p < 10000$.