  A characterization of $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamard
matrices is described, depending on the notions of {\em distributions}, {\em
ingredients} and {\em recipes}. In particular, these notions lead to the
establishment of some bounds on the number and distribution of 2-coboundaries
over $\mathbb{Z}_t \times \mathbb{Z} _2^2$ to use and the way in which they
have to be combined in order to obtain a $\mathbb{Z} _t \times
\mathbb{Z}_2^2$-cocyclic Hadamard matrix. Exhaustive searches have been
performed, so that the table in p. 132 in [4] is corrected and completed.
Furthermore, we identify four different operations on the set of coboundaries
defining $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic matrices, which
preserve orthogonality. We split the set of Hadamard matrices into disjoint
orbits, define representatives for them and take advantage of this fact to
compute them in an easier way than the usual purely exhaustive way, in terms of
{\em diagrams}. Let ${\cal H}$ be the set of cocyclic Hadamard matrices over
$\mathbb{Z}_t \times \mathbb{Z}_2^2$ having a symmetric diagram. We also prove
that the set of Williamson type matrices is a subset of ${\cal H}$ of size
$\frac{|{\cal H}|}{t}$.
