  If $G$ is a centreless group, then $\tau(G)$ denotes the height of the
automorphism tower of $G$. We prove that it is consistent that for every
cardinal $\lambda$ and every ordinal $\alpha < \lambda$, there exists a
centreless group $G$ such that (a) $\tau(G) = \alpha$; and (b) if $\beta$ is
any ordinal such that $1 \leq \beta < \lambda$, then there exists a notion of
forcing $P$, which preserves cofinalities and cardinalities, such that $\tau(G)
= \beta$ in the corresponding generic extension $V^{P}$.
