  This paper presents algorithmic approaches to study superspecial
hyperelliptic curves. The algorithms proposed in this paper are: an algorithm
to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields
$\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not
necessarily superspecial) hyperelliptic curve over finite fields. The first
algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and
$q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of
genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over
$\mathbb{F}_{p^2}$ for $11 \leq p \leq 19$ with our implementation on a
computer algebra system Magma. Moreover, we found maximal hyperelliptic curves
and minimal hyperelliptic curves over $\mathbb{F}_{p^2}$ from among enumerated
superspecial ones. The second algorithm computes an automorphism as a concrete
element in (a quotient of) a linear group in the general linear group of degree
$2$.
