  This paper was motivated by a remarkable group, the maximal subgroup
$M=S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}\ltimes2^{6+1}_{-}$ of the sporadic
simple group ${\rm Fi}_{23}$, where $S_3$ is the symmetric group of degree 3,
and $2^{2+1}_{-}$, $3^{2+1}$ and $2^{6+1}_{-}$ denote extraspecial groups. The
representation $3^{2+1}\to{\rm GL}(3,\mathbb{F}_4)\to{\rm GL}(6,\mathbb{F}_2)$
extends (remarkably) to $S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}$ and preserves a
quadratic form (of minus type) which allows the construction of $M$. The paper
describes certain (Weil) representations of extraspecial groups which extend,
and preserve various forms. Incidentally, $M$ is a remarkable solvable group
with derived length 10, and composition length 24.
