  Let $F$ be a field with characteristic $\neq 2$. We show that $F$ is a
nonrigid field if and only if certain small 2-groups occur as Galois groups
over $F$. These results provide new "automatic realizability" results for
Galois groups over $F$. The groups we consider demonstrate the inequality of
two particular metabelian 2-extensions of $F$ which are unequal precisely when
$F$ is a nonrigid field. Using known results on connections between rigidity
and existence of certain valuations, we obtain Galois-theoretic criteria for
the existence of these valuations.
