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A characterization of $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamardmatrices is described, depending on the notions of {\em distributions}, {\emingredients} and {\em recipes}. In particular, these notions lead to theestablishment of some bounds on the number and distribution of 2-coboundariesover $\mathbb{Z}_t \times \mathbb{Z} _2^2$ to use and the way in which theyhave to be combined in order to obtain a $\mathbb{Z} _t \times\mathbb{Z}_2^2$-cocyclic Hadamard matrix. Exhaustive searches have beenperformed, so that the table in p. 132 in [4] is corrected and completed.Furthermore, we identify four different operations on the set of coboundariesdefining $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic matrices, whichpreserve orthogonality. We split the set of Hadamard matrices into disjointorbits, define representatives for them and take advantage of this fact tocompute 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 provethat the set of Williamson type matrices is a subset of ${\cal H}$ of size$\frac{|{\cal H}|}{t}$.

On $\mathbb{Z}_t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices