  The non-abelian groups with abelian group of automorphisms are widely
studied. Following Earnley, such groups are called Miller groups, since the
first example of such a group was given by Miller in 1913. Many other examples
of Miller $p$-groups have been constructed by several authors. Recently, A.
Caranti [{\it Israel J. Mathematics {\bf 205} (2015), 235-246}] provided module
theoretic methods for constructing non-special Miller $p$-groups from special
Miller $p$-groups. By constructing examples, we show that these methods do not
always work. We also provide a sufficient condition on special Miller $p$-group
for which the methods of Caranti work.
