The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let
G be the direct product of a generalized extraspecial ?-group
E and a free abelian group
A with rank
m, where
$$E = \left\{ {\left( {\begin{array}{*{20}{c}}1&{k{\alpha _1}}&{k{\alpha _2}}& \cdots &{k{\alpha _n}}&{{\alpha _{n + 1}}} \\0&1&0& \cdots &0&{{\alpha _{n + 2}}} \\\vdots & \vdots & \vdots &{}& \vdots & \vdots \\0&0&0& \cdots &1&{{\alpha _{2n + 1}}} \\0&0&0& \cdots &0&1\end{array}} \right)\left| {{\alpha _i} \in Z,i = 1,2,...,2n + 1} \right.} \right\},$$
where
k is a positive integer. Let Aut
G′G be the normal subgroup of Aut
G consisting of all elements of Aut
G which act trivially on the derived subgroup
G′ of
G, and Aut
G/ζG,ζG G be the normal subgroup of Aut
G consisting of all central automorphisms of
G which also act trivially on the center
ζG of
G. Then (i) The extension 1 → Aut
G′G → Aut
G → Aut
G′ → 1 is split. (ii) Aut
G′G/Aut
G/ζG,ζG G ? Sp(2
n,
Z) × (GL(
m,
Z) × (?)
m ). (iii) Aut
G/ζG,ζG G/Inn
G ? (?
k )
2n ⊕ (?
k )
2nm.