  Let $p$ be a prime. The $2$-primary part of the class group of the pure
quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and
Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the
known results in the case $p\equiv \pm 1\bmod 16$. In particular, we determine
all primes $p$ such that $4$ does not divide the class number of
$\mathbb{Q}(\sqrt[4]{p})$. We also conjecture a relation between the class
numbers of $\mathbb{Q}(\sqrt[4]{p})$ and $\mathbb{Q}(\sqrt{-2p})$. We show that
this conjecture implies a distribution result of the $2$-class numbers of
$\mathbb{Q}(\sqrt[4]{p})$.
