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We determine the order of the automorphism group Aut 
$(W)$ for each member 
$W$ of an important family of finite 
$p$ -groups that may be constructed as iterated regular wreath products of cyclic groups. We use a method based on representation theory.
We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group 
$G$ on a finite vector space 
$V$ under certain conditions, to a more general class of finite solvable groups $G$ . This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.
