  We construct, for the first time, various types of specific non-special
finite $p$-groups having abelian automorphism group. More specifically, we
construct groups $G$ with abelian automorphism group such that $\gamma_2(G) <
\mathrm{Z}(G) < \Phi(G)$, where $\gamma_2(G)$, $\mathrm{Z}(G)$ and $\Phi(G)$
denote the commutator subgroup, the center and the Frattini subgroup of $G$
respectively. For a finite $p$-group $G$ with elementary abelian automorphism
group, we show that at least one of the following two conditions holds true:
(i) $\mathrm{Z}(G) = \Phi(G)$ is elementary abelian; (ii) $\gamma_2(G) =
\Phi(G)$ is elementary abelian, where $p$ is an odd prime. We construct
examples to show the existence of groups $G$ with elementary abelian
automorphism group for which exactly one of the above two conditions holds
true.
