The automorphism group of finite 2-groups associated to the Macdonald group

We consider the Macdonald group [Formula: see text] and its Sylow 2-subgroup [Formula: see text], where [Formula: see text] and [Formula: see text] is odd. Then J has order [Formula: see text], and nilpotency class 5 if [Formula: see text] and 3 if [Formula: see text]. We determine the automorphism group of the 2-groups J, [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. Explicit multiplication, power, and commutator formulas for J, H and K are given, and used in the calculation of [Formula: see text], [Formula: see text] and [Formula: see text].