  There are several examples in the literature of finite non-abelian $p$-groups
whose automorphism group is abelian. For some time only examples that were
special $p$-groups were known, until Jain and Yadav [JY12] and Jain, Rai and
Yadav [JRY13] constructed several non-special examples. In this paper we show
how a simple module-theoretic approach allows the construction of non-special
examples, starting from special ones constructed by several authors, while at
the same time avoiding further direct calculations.
