Coleman automorphisms of generalized dihedral groups

Let D be a generalized dihedral group and AutCol(D) its Coleman automorphism group. Denote by OutCol(D) the quotient group of AutCol(D) by Inn(D), where Inn(D) is the inner automorphism group of D. It is proved that either OutCol(D) = 1 or OutCol(D) is an elementary abelian 2-group whose order is completely determined by the cardinality of .(D). Furthermore, a necessary and sufficient condition for OutCol(D) = 1 is obtained. In addition, whenever OutCol(D) . 1, it is proved that AutCol(D) is a split extension of Inn(D) by an elementary abelian 2-group for which an explicit description is given.