  We recall the notion of a quadratic form parameter $Q$ over the integers and
of extended quadratic forms with values in $Q$, which we call $Q$-forms.
Certain form parameters $Q$ appeared in Wall's work on the classification of
almost closed $(q{-}1)$-connected $2q$-manifolds via $Q$-forms. The algebraic
theory of general form parameters and extended quadratic forms has been studied
in a variety of far more general settings by authors including Bak, Baues,
Ranicki and Schlichting.
  In this paper we classify all quadratic form parameters $Q$ over the
integers, determine the category of quadratic form parameters $\mathbf{FP}$ and
compute the Witt group functor, \[ W_0 \colon \mathbf{FP} \to \mathbf{Ab},
\quad Q \mapsto W_0(Q),\] where $\mathbf{Ab}$ is the category of finitely
generated abelian groups and $W_0(Q)$ is the Witt group of nonsingular
$Q$-forms.
